Probably every business would love to dominate their product category.

The textbook case is Apple. They so dominate several fields - portable music players, tablet computers, smartphones - that you can't have an intelligent discussion without reference to iPod, iPad, or iPhone.

How did they get that way (besides the fact that Steve Jobs is a legitimate, screaming genius)?

Here's an excellent article on the subject:

Why Apple's Distortion Field Works

Notice that in several cases, innovations that Apple is credited with were NOT in fact first introduced by them.