In the 19th century, many things started to change. The world of 1800 would not be a shock to a person from the 15th century. But for some reason, the world of today would be an incredible shock to someone from the 19th century. We have airplanes, automobiles, high speed trains. We've even put someone on the moon. All this happened in approximately 150 years, from the start of the Industrial Revolution until now. This is all fine and dandy, but why did the Industrial revolution start when it did? What happened?
It all started with a fairly simple change in the way that crops were managed and produced in England. In the late 1700's farmers began to realize that rather than leaving their fields fallow after a harvest, they could plant beans or other products that resulted in restoring the fertility of the soil. Then as they began to rotate their crops in different fields, the crop production began to increase. As crop production began to increase, the food supplies for the general population as well as livesock increased, so both began to increase. The sale of increasing numbers of livestock and volumes of crops resulted in the need for better routes of transportation, so water canals were developed and roads were improved. Eventually the railroad came along and helped spread these new growths in commerce.
Ultimately, each small change led to a new need which was then addressed by another change. Sometimes it was as simple as rotating crops, and other times it involved the inventions of new equipment that could more efficiently harvest the crops or transport them once they were harvested. The successful increase in the volume of crops produced ultimately led to an increasing demand for better ways to process the crops.
As a result of this agricultural change, commerce within the country began to increase as well. Suddenly there were more people who were able to earn wages that allowed them to have disposable income. This resulted in the need for other goods to sell to this increasing working class, and that led to more improvements to help mass produce products, such as textiles. From there it just continued to snowball.
So, having said all that, the simple answer is that a simple discovery of a way to rotate crops to keep the soil more fertile started the whole thing rolling!
The industrial revolution occurred as a result of scientific advances in Europe. Specifically the Steam Engine and related Manufacturing Technology. However the Industrial Revolution began specifically in Britain, since wages in Britain were significantly higher than on the continent. This disparity increased the incentive for British businessmen (vs continental European) to invest in labor saving machines, which began the Industrial Revolution.
Unprecedented scientific advancements allowed economics to favor mechanization and spawn the Industrial Revolution.
The knowledge necessary to build the machines that backed the industrial revolution (notably, steam engines) had been around for quite a while before that (ancient Greeks were fully aware of how to use steam to move things); but it had never been used widely for a variety of reasons:
In the short run, human labour is vastly cheaper than designing, building, running and mantaining machines, which instead leads to much greater efficiency in the long run. What actually started the whole thing were changing social and economic factors, which made an industrial economy worthwhile.
It's a classic question of why the industrial revolution happened when and where it did. And answers range to everything from "Good Cannon makers" to "Because the British drank tea!"
It's likely that the question is impossible to answer conclusively. But what I would like to do is to set up a number of prerequisites for an industrial revolution. It's not certain that an industrial revolution will happen when these prerequisites are met, but I do think they are required, and I also think the 18th century Britain is the first time in history where they all have been fulfilled at the same time.
1. High agricultural yield, leading to cheap labour
This is what Steven Drennon talks about in his answer. The British Agricultural revolution led to farms being able to produce more food and feed more people. As a result, the population in cities increase, and there is a lot of labour available.
2. Technological ability
The ancient Greeks invented steam-powered toys. Why didn't they invent a steam engine? Because they didn't have the technology. The first attempts at steam engines in Britain and France failed, because the materials weren't up to the job of handling steam pressure. Also Watt could not have invented the condenser, and made the steam engine a much more economical alternative, unless he had understood boiling and condensing and how related to the different temperatures in the machine.
3. A capitalist economic system
In addition to a work force, you also need inventors who invent, industrialists who run things and investors that provide the money.
Point 1 and 2 in turn has a common prerequisite:
4. A scientific method
You can't get high crop yields if you think the most important part of the crop yield is praying to gods or making fertilizer by burying rams horns by the full moon. You will not learn the laws of thermodynamics if you think that converting water to steam is a form of magic.
And all of the above have a last prerequisite:
1. and 3. needs economic liberty; private ownership of land and other means of production, and the right to do what you think is best to make money and invest that money.
2. and 4. needs liberty in ideology and thinking. The liberty to say that earth may not be in the centre of a theological universe, but zip around all by itself in a universe ruled by mechanics.
3. Also has a final prerequisite:
5. A big trading network
So there are people that can accumulate wealth that can be invested in industry.
There is also an argument that you need a good source of energy, ie coal. And that aragument is that late medieval and renaissance Netherlands had pretty much all of the liberty requirements fullfilled, but still did not get an agricultural revolution. But perhaps it was missing the natural resources needed. On the other hand I have also seen the argument that it simply was too early, the science and technology wasn't there yet, when the Netherlands had liberty and money.
This allowed for the substitution of fuel as sources of energy (coal generated steam or steam engines, oil drove internal combustion engines) on a much larger scale than human or animal power. In essence, human beings were freed up to manage high-powered "engines" (and "machines" driven by engines) in factories, instead of having to put up a large part of the labor power on a much smaller scale.
Accompanying these changes was the development of banking, which allowed many "small" deposits to be combined into loans for the devlopment of factories and other large scale enterprises, to corporations that allowed people to invest in these enterprises with "limited liability."
This sequence is best represented by the industrialization branch of the game Civilization II, from steam engine, to railroads, industrialization, the corporation, (oil) refining, and (internal) combustion.