Question is, how does the system know that long string of characters is tied to me? I didn't sign up for an account, so there is no username and password to tied that string to me.
That super-long string is called the Bitcoin Address. It is a human-readable representation of the public key used to accept the money. The Bitcoin client generates the key pair for you, and each key pair is unique. No other client will have the same key pair, so only you can claim the money sent to your Bitcoin address.
And what if I changed computers and downloaded a new client? The string will be different again.
Rule #1 of using bitcoin: protect your wallet. If you lose it or overwrite it, all your money is lost to you. Kind of like losing your physical wallet. The wallet is the file wallet.dat.
And keep in mind that the address will
change each time you accept money. The client can show you a list of all the Bitcoin addresses you have in your wallet.