There is a balanceOf function, but it only displays one uint (the token identifier) I see most ERC721 tokens doing this so I am confused on how to view all owned tokens.
Per https://github.com/ethereum/EIPs/blob/master/EIPS/eip-721.md, there's an optional "enumeration extension". If this is implemented for the token you're interested in, then you can just call
balanceOf to get the number of tokens owned by the account, followed by
tokenOfOwnerByIndex in a loop to get each owned token ID.
If it isn't implemented, then there's no well-defined way to discover all of a user's owned tokens, but there may be a non-standard way to do it for the specific token you're interested in.
balanceOf actually returns the number of owned tokens and not the identifier of an owned token.
Therefore I'd assume the process could go something like this:
1) Find out how many tokens an address has (
balanceOf). If this is more than zero, continue the process.
2) Find out which tokens exist in the contract. The standard offers no direct functionality for this so this depends on the implementation. Or, as is probably the case, simply start enumerating from zero upwards hoping that the tokens are enumerable in that way (and not assigned for example to random numbers). So you would check "does someone own token 0? does someone own token 1? .."
3) Once you have found all the tokens owned by the address you can stop the process.
There is basically no way to easily view all owned tokens. My Ethereum wallets own probably lots of random worthless tokens which someone has airdropped and I don't even know about it.
Some services (wallets) display many of the owned tokens. I'm unsure how exactly they do that but most likely they simply have a database of known token contracts and they check whether a certain address owns those known tokens.
If I now created a new token and donated some of it to random wallets nobody would probably know about it unless I somehow tell wallet creators "please include my token contract in your checks".
At least myetherwallet used to display all (known) ERC20 tokens still about a year ago but the quickly they gave up and last I checked you had to manually enter the contract address to see those tokens. The amount of different token contracts exploded a year ago and it was probably too much hassle to try to keep track of them (to store them all in their database).