If that coin is one of the inputs to a later valid transaction, then it is spent and cannot be spent again.
That later transaction is the record that somebody spent the coin; it is signed with the private key, and that key (which Bitcoin keeps for you in your wallet) should be known only to the owner.
"That later transaction is on the recor that somebody spent the coin; it is signed with the private key..."
So, the chain contains a full history of transactions, and a coin's current owner is determined by all nodes as being the last signature in the chain associated with a coin. Am I reading this right?