Oh man this is a way more complicated question than you probably realize. Hundreds of books have been written about this.

Basically cotton was in use in warmer environments for clothes for an extremely long time. Cotton simply couldn't grow particularly well in colder environments, and before long distance trading routes you couldn't bring it to Europe in bulk. Hence most clothes in Europe were made of linen (made from flax) or wool (from sheep).

This started to change in the 1600s. The European East India Companies (primarily the Dutch, French, and English) started importing cotton clothe from India. It quickly became fashionable, and started to become a major import, and what's more, a strong competitor to domestic linen and wool. While domestic interests tried to get it banned in European countries, it quickly became the fabric of choice for European fashions, especially since Indian cloth (callicoes) were dyed with interesting patterns.

Once the trade was effectively banned in the early 1700s in England, domestic producers looked for ways to reproduce the cloth using imported raw cotton. They had a convenient source in the American South, and what's more there was a series of supporting innovations in cotton manufacture. Since cotton is much easier to work with on an industrial scale than wool or linen (it has longer fibers and grows much more easily and productively than flax or wool) it came to dominate the growing textile manufacture.

One thing leads to another and everyone everywhere is wearing British made cotton clothing. The same techniques to mass produce cotton cloth are applied to a variety of other industries, and voilà! you have yourself an industrial revolution.