Or, it means that when you create the transaction that spends Alice's coins and pays them to Bob, you also create and include the unique attribute that qualifies the transaction to do so. The transaction to Charlie or anyone else will not have that unique attribute so the network will know they are invalid if they see them.
You have two possible cases:
1) Alice can send coins to Bob, Charlie or someone else
2) Alice cannot send coins to Bob, Charlie or someone else, she is forced to move coins to some predefined destination
In the first case, it is technically possible to create valid transaction "Alice->Bob", as well as "Alice->Charlie" and "Alice->Daniel". If you have all transactions, you can prove that "Alice->Bob" is included in the chain, so nothing else like "Alice->Charlie" or "Alice->Daniel" can be included later. To do that, you have to "be aware of all transactions", as Satoshi wrote.
In the second case, the coin is useless, because there is no owner of any coin. The whole history is predefined, because for each coin there is only one valid transaction, so the only one valid list of inputs and outputs, and the only one valid list of amounts, everything has to be known when the coin is created. Then you can look at Alice's coin of 50 BTC and you know that moving 10 BTC to Bob and 40 BTC to Charlie is the only valid transaction that you can create. Everything is predefined, so no blockchain is needed, but the whole system is then useless.