ERC-20: For money and money-like tokens.
ERC-721: For things and thing-like tokens.
These are contract INTERFACES. When someone says they have an “ERC-20” token that just means that the token contract responds to a common set of commands. That’s all. It means that the token can be transferred, requested, approved, etc in the same way as any other ERC-20 tokens. It does NOT mean the token has value. It does NOT mean the token actually does anything interesting. In fact, by itself it means almost nothing.
An interface defines WHAT something does, not how it does it. ERC-20 is an interface. An ERC-20 token can have the interface and still do basically nothing. This can’t be said enough: Having an ERC-20 token is MEANINGLESS. If someone asks for money for their ERC-20 token and the best reason they can do is that they have an Eth token, ignore them.
ERC-721 is a more advanced token that is “non-identical”.
Think of ERC-20 as the token type for things that are money (any $5 bill is worth the same as any other $5 bill, usually) and ERC-721 as the token type for collectibles. Like every $10 note has same value and you can pay a $10 bill with any $10 note in the world.
In ERC-721, each token is completely unique and non-interchangeable with other tokens. Another example: Pets. Many people have dogs, but THAT dog is theirs and they will not accept some other dog as a substitute. You can use ERC-721 tokens to represent those dogs and dog ownership.
An ERC-721 can be record of your criminal history and it will always be yours, you can’t exchange your criminal history with someone else, It can be linked to a property and whoever will have the token linked to a certain property will be the owner of that property.
Each token is it’s own mini-database of who owns what. The “token” is just an entry in the token contract, and who “owns” a token is recorded in the contract. A token is never in “your” wallet.
If you have the private key of the wallet whose public key is the owner of certain token than you’re the owner of that token.