How do I switch sides in the Civil War?

If you've not yet completed The Jagged Crown, you can turn the crown in to Ulfric Stormcloak in the Palace of the Kings in Windhelm to switch sides to the Stormcloaks.

Those that originally sided with the Stormcloaks, at this point can also switch sides to the Imperial Legion and give the crown to General Tullius in Castle Dour in Solitude.

If you're past that point, it's too late to switch.

It depends how far along the quest line you are.

There is a quest called The Jagged Crown where you have to retrieve.... drum roll... the Jagged Crown

Instead of taking it to the quest giver, you can take it to Ulfric Stormcloak to swap allegiance. I believe this is the only time you can do so.

This also works if you want to swap from Stormcloaks to Imperials.