I'm trying to create NFTs (non-fungible tokens), but I don't quite understand this part - is each token its own contract? or are all of them stored in the NFT equivalent of a 'balances' mapping in a factory contract? If these are both options which is better?
No, NFTS are not contracts of their own.
A single NFT is a
uint256 ID, which is mapped to an owner address. In solidity, this would be something like
mapping(uint => address) tokenIds
The spec lists a number of reference implementations, you should probably use one of those rather than roll your own.