What do historians mean when they talk about "supply side" and "demand side" explanations of the industrial revolution?

There is a historical question about why the industrial revolution happened in the late 1700s in Great Britain, rather than a different time or place. Some explanations involve either the supply of innovations, or innovations in the supply of products. For instance, maybe Britain had a better patent system, making it safer to innovate and leading to more innovations. This is a change on the supply side of the economy - inventors were encouraged, so they invented more and were able to supply more stuff. (This is an explanation I basically made up as an example of a supply side explanation. I don't really think it's right.)

Alternatively, maybe Britain had a particular need for steam power or mechanical looms, so that any innovations made there would actually take off. Maybe other countries were just as ready to innovate, but the innovations there weren't as profitable. That is a demand side explanation. The way I put it, demand side explanations sound less plausible, but this article in the Economist gives case for a realistic demand side explanation.

From the abstract to Mokyr's article, it is clear that he supports a supply side explanation, contrary to the more recent book referenced in the Economist article. So this seems to be still a live debate.

Of course, the supply and demand sides are general categories in economic theory, but this is how they are being applied in this context.

Supply side is

The specific set of foolish ideas that has laid claim to the name "supply­ side economics" is a crank doctrine that would have had little influ­ence if it did not appeal to the prejudices of editors and wealthy men. But over the past few decades there has been a steady drift in emphasis in economic thinking away from the demand side to the supply side of the economy. (p182)
_{Paul Krugman: "The Return of Depression Economics and the Crisis of 2008", WW Norton: New York, 2009.}

Paul Krugman: "The Tax-Cut Zombies", New York Times, Dec. 23, 2005

Mr. David Stockman has said that supply-side economics was merely a cover for the trickle-down approach to economic policy—what an older and less elegant generation called the horse-and-sparrow theory: If you feed the horse enough oats, some will pass through to the road for the sparrows.
_{John Kenneth Galbraith: "Recession Economics", The New York Review of Books, Feb 2, 1982}

In other words, supply-side is:

The model we present is a highly simplified one. We call our two factors of production capital and labor, but we do not distinguish one as fixed and the other as variable. Since the model is static, we do not attempt to analyze the process of capital formation. Instead, we assume that at any point there exist fixed stocks of capital and labor and that these stocks must be allocated either to household production or to market-sector production.

_{Victor A. Canto & Douglas H. Joines & Arthur B. Laffer: "Foundations of supply-side economics. Theory and Evidence", Academic Press: New York, London, 1983, p1.}

While demand-side is:

Demand-side economics is a macroeconomic theory which argues that economic growth is most effectively created by high demand for products and services. According to demand-side economics, output is determined by effective demand. High consumer spending leads to business expansion resulting in greater employment opportunities. Higher levels of employment creates a multiplier effect that further stimulates aggregate demand leading to greater economic growth. Demand-side economists argue tax breaks for the wealthy produce little, if any, economic benefit because most of the additional money is not spent on goods or services. Instead, they argue increased government spending will help to grow the economy by spurring additional employment opportunities.
British economist John Maynard Keynes is the most celebrated of demand-side economic theorists. Keynes saw his theories successfully demonstrated in the 1930s when they helped to end the Great Depression and into the 1950s and 60s when capitalism experienced its Golden Age. Additional proponents of demand-side economics include Leon Keyserling, John Kenneth Galbraith, Hyman Minsky, Joseph Stiglitz, James K. Galbraith, Steve Keen and Nouriel Roubini. _(WP)

To understand what Keynes had in mind, imagine federal budget policies that would increase aggregate demand in Exhibit 6, shifting the aggregate demand curve to the right, back to its original position. Such a shift would raise real GDP, which would increase employment. According to the Keynesian prescription, the miracle drug of fiscal policy—changes in government spending and taxes—could compensate for what he viewed as the instability of private-sector spending, especially investment. If demand in the private sector declined, Keynes said the government should pick up the slack. We can think of the Keynesian approach as demand-side economics because it focused on how changes in aggregate demand might promote full employment. Keynes argued that government stimulus could shock the economy out of its depression. Once investment returned to normal levels, and the economy started growing on its own, the government’s shock treatment would no longer be necessary.
_{William A. McEachern: "Economics: A Contemporary Introduction" South-Western College Pub, Year: ⁹2011.}

Meaning that both theories alone or an undue emphasis on one of them is the result of idiotic ideology taking over sound reasoning. Letting supply-side govern economic development leads into crises like Great Depression and keeps the economy in there for a longer time. Unfortunately, favouring one side over the other is often coloured by or the result of firm biases of writers. Conservatives going for supply, social-democrat-like (US: liberals) going for the demand-side approach.

This is evident from looking at an economics textbook for beginners:

Learning Goals
After reading this chapter, you will be able to:
■ Understand how the components of the standard trade model, production possibilities frontiers, isovalue lines, and indifference curves fit together to illustrate how trade patterns are established by a combination of supply-side and demand-side factors.

_{Paul R. Krugman & Maurice Obstfeld & Marc J. Melitz: "International Economics. Theory & Policy", Pearson, 2018, p152.}

So you need not only a free market (supply-side) but also government guidance and regulations as supreme arbiter to keep in balance with the demand-side as otherwise all those crises Marx predicted come crashing down undampened on the economy.

Or, as Adam Smith put it:

A landlord, a farmer, a master manufacturer, a merchant, though they did not employ a single workman, could generally live a year or two upon the stocks which they have already acquired. Many workmen could not subsist a week, few could subsist a month, and scarce any a year without employment. In the long run, the workman may be as necessary to his master as his master is to him, but the necessity is not so immediate.
_{(Adam Smith: "Wealth of Nations" (1776) Book I, ch 8)}

One example in trying to apply this to the Industrial Revolution:

If demand was not a "factor," what exactly was its place in the industrial revolution? To start with, we observe that any supply shift will affect the economy in direct proportion to the proportional size of the industry affected relative to the economy as a whole. For a once and for all fall in costs the demand structure only matters for determining the composition of the increment in national income but not its size. If the supply shift is a continuous process, however, the shape of the demand curve does matter, because it determines the future pattern of the relative size of the industry in question in the economy. To be more precise: if the own price elasticity is more than unity in absolute value it can be shown that (1) the overall impact of a constant rate of cost reduction will increase over time, and (2) the spillover effects for all other goods taken together are negative. This casts a peculiar light on the role of so-called leading sectors which are supposed to grow due to a very elastic demand curve. Moreover, as output expands, the economy will move down into the inelastic segment of the demand curve. At that point the continuous fall in costs will result in increased demand for all goods (that is, for the good in question and for all other goods taken together), but this impact will slowly peter out over time. Something similar to this process happened in the market for textiles between 1760 and 1860. The elastic demand caused output to grow very rapidly as a response to initial price reductions, so that subsequent price reductions were applied to a much wider base.
Secondly, there is the stability of demand. While the absolute level of demand for industrial goods cannot be used to explain the timing and speed of the industrialization process, heavy fluctuations in de- mand had an adverse effect on growth, due to the substantial costs of resource reallocation, the acquisition of new information, and so forth. Wars, revolutions, blockades, tariffs, harvest failures, and other unanticipated catastrophes inhibited growth not so much through the first moments of the demand function parameters as through the second moments.
A third way in which demand related factors could have been important in determining the speed and timing of the industrialization process is through their determination of the intersectoral terms of trade. As a consequence of a change in the terms of trade income distribution may change. For example, income may be redistributed from industrial workers and capitalists to landowners. It is, in fact, possible that industrialization could lead to "immiseration" of the modern sector in a way that is analogous to the well-known possibility of "immiseration" of a country increasing its exports. Note, however, that "immiserizing growth" can occur at the level of the economy but not at the level of the entire world. Similarly in a closed economy, it can occur at the level of a sector, but the economy as a whole is better off. Still, if the modern sector has a higher savings propensity or lower risk averseness than the other sector, such a worsening of the terms of trade may affect the rate of growth.
_{Joel Mokyr: "Demand vs. Supply in the Industrial Revolution", The Journal of Economic History, Vol. 37, No. 4 (Dec., 1977), pp. 981-1008.}

But this is very much contested by for example

Robert Allen's analysis will delight many economists, for he deals in measurable factors such as wages and prices. An American professor of economic history at Oxford University and long a writer in this field, he suggests that most explanations for Britain's industrial revolution focus too much on supply—of inquiring scientists, landless workers, helpful laws. These conditions were conducive to a great leap forward but not sufficient. Nor were they exclusive to Britain. Property rights were arguably more secure in France; much of the science behind the steam engine took place in Italy and Germany; the Dutch were highly urbanised. The industrial revolution occurred in Britain in the 18th and early 19th centuries for one overwhelming reason, he argues: it was profitable there and then. It met a demand.
"The industrial revolution. Supply and demand. Why Britain got there first" Economist, May 21st 2009.

Referencing Allen: "The British Industrial Revolution in Global Perspective", Cambridge University Press: Cambridge, 2009.

The industrious revolution was a process of household-based resource reallocation that increased both the supply of marketed commodities and labor and the demand for market-supplied goods. The industrious revolution was a household-level change with important demand-side features that preceded the Industrial Revolution, a supply-side phenomenon. It has implications for nineteenth- and twentieth-century economic history.

One interpretation of the new industrious household-one to which I give assent in my pessimistic moments-is that the absorption into the market economy of the last remaining substance of the household represents a final frontier of capitalism, that it demonstrates the truth of Schumpeter's famous observation about the bourgeois family: "The capitalist order rests on props made of extra-capitalist material [and] derives its energy from extra-capitalist patterns of behavior which at the same time it is bound to destroy."

_{Jan De Vries: "The Industrial Revolution and the Industrious Revolution ", The Journal of Economic History, Vol. 54, No. 2, 1994, pp. 249-270. (jstor)}

And all this is so much coloured by the needs and goals of the present, if it's not historians but economists who write the narrative:

'Even if … the history of "the" Industrial Revolution is a "thrice squeezed orange", there remains an astonishing amount of juice in it.'

In Section I, the discussion is set within the framework of Cannadine's hypothesis that studies of the Industrial Revolution are heavily influenced by contemporary events (thereby potentially explaining the repeated orange squeezings as times change). It is argued here that this view needs to be developed further, especially to take account of the changing input from economic theory. In this context, the 'consumerist approach' associated with McKendrick is understood as a reaction against the current orthodoxy, which increasingly embraces cliometrics. Revisionism occurs, partly as a switch of emphasis from supply to demand, partly as a reinterpretation of the Industrial Revolution as the birth pangs of modern consumer society, and partly as a means of retaining or opening up explanatory factors beyond those within the narrowing confines of orthodox economic analysis.
_{Ben Fine & Ellen Leopold: "Consumerism and the Industrial Revolution", Social History, Vol. 15, No. 2 (May, 1990), pp. 151-179, (jstor)}

For most of history, the economy was supply constrained. That is, people could manufacture goods at a slower rate than they were able to consume them. So if you could make more or better "mousetraps," you could probably find a buyer for them. Under these circumstances, the important variable to monitor was the supply of goods. In an economy of "scarcity" any additional supply would create its own demand. This was the premise of the so-called Say's Law, which "described" the world up to the Industrial Revolution, although it came later. "Supply," in this context, refers to "aggregate" supply in the whole economy of say, the United States, if not the whole world.

During the late 19th and early 20th centuries, there arose the phenomenon of mass production. Now it was possible to manufacture goods in large quantity, sometimes to "excess." In this "new" era, it was important for manufacturers to restrict the quantity of a given type of goods produced, and not "flood the market." (This had never been a problem under the old, supply-based regime.)

When there is excess capacity in many parts of the markets, (as occurred from time to time in the middle and late 20th century and later), not only companies, but governments have to get into the act, e.g by raising interest rates or taxes to discourage too much spending, or "excess demand." This process is called "demand management," and ultimately refers to "aggregate" demand (not just in one good or industry).