Hooray, paperwork! Luckily, the PCT forms aren't too painful. Here's a brief overview, although the PCTA website is a great resource that covers everything in more detail.
Short Hikes on the PCT
If you're doing a PCT hike less than 500 miles, you'll need to fill out a permit with the specific agency that manages the trailhead you're starting at. If that agency doesn't require a permit, get one from the first agency that requires one on your itinerary. Your permit will cover you for the entire trip, even if you cross management boundaries. Get on this as soon as you decide to hike, because some places have quotas.
PCTA Long-Distance Hiking Permit
If your PCT hike is longer than 500 miles, you can get a permit from the PCTA. The permit is free, although add $15 if you'd like to summit Mt. Whitney. They also give you the option of becoming a member of the PCTA, which seems like a solid thing to do (karma and all.) You apply online, but you also need to send them a signed copy via mail, fax, or email. Give them 2-3 weeks to process everything. You'll get a physical copy of your permit in the mail.
California Camping Permit
You'll also need a California Campfire Permit to use a stove, lantern, or campfire outside a developed campground or recreation area. They are free and valid for one calendar year -- all you need to do is complete a simple online quiz and try not to burn the forest down!
(POST HIKE NOTE: My permits were checked by rangers twice during my trip -- once in the Yosemite backcountry and once heading down from Mt. Whitney. It's not just a formality. Hang on to them!)