They'll replace it at guest services at any of the main parks. I believe they can charge $10, but the only time we lost a pass they replaced it for free.
They just ask for ID. We had the other passes for our party (my husband somehow managed to throw his away at AK). It was painless, and if you have the other passes I'm sure you'll have no problem at all.
One of ours accidently got thrown into the wash, and came out whistle clean -absolutely nothing was on it. They replaced it--you are in their computer. You merely need proof of who you are. And they will invalidate the other one in case it's found.
The procedures mentioned above only work for already activated annual passes. if you have lost a voucher, you will need the numbers off the back of the voucher.
It's also a good idea to make a photo copy of the backs of all your passes. It may not be that beneficial for lost APs, but it may save your bacon if you lose a Park Hopper. I keep a copy of ours back at the hotel room.