As far as we know, riding side saddle was invented in Medieval Europe to allow women to ride in long skirts. (It wasn't considered appropriate to wear trousers and in many cases the woman would be trying to get somewhere where she had to look good).
The earliest side saddles didn't really allow women to ride - she sat sideways and her mount was led or ponied by a man. The proper side saddle wasn't developed until the 16th century and the modern side saddle, in which a woman could do anything a man riding astride could do, came into existence with the invention of the leaping head in the 1830s.
Of course, some Medieval women chose to ride astride so they could keep up with the men. There's a famous portrait of Catherine the Great riding astride in a cavalry uniform.
Nowadays, almost all women ride astride, but side saddle classes are often seen at shows. Queen Elizabeth II rode side saddle on formal occasions (but preferred astride in private). Women often ride side saddle in historical reenactments.
Modern side saddle rider at a horse show in Ireland. Source: Declan via Wikimedia commons.
I once saw a woman do ring jousting side saddle...it was quite a display of horsemanship. Some women also find side saddle more comfortable and it can sometimes be safer (harder to fall off, but with a greater risk of injury if the horse falls). A few men also ride side saddle - it can be more comfortable if you have a bad back. And side saddles are sometimes used by disabled riders, especially individuals who have lost part of a leg.