I just saw one of the eos accounts has a "claim" key. What is the claim key for and how is it different from the rest?
Theoretically you can create as many key as you want and as many permission as you want.
A single key pair can be related to multiple permission. Moreover you can specify a permission name and link a permission to a specified action in a smart contract.
The 'claim' key you said is a key (permission) created by the owner of the account, it's just a name. Active and owner, on the other hand, are mandatory and have fixed name.
You can find an awesome video (IMO) that talks about permission, on youtube, in the EOS Canada channel.