Because a wallet consists of addresses and each of those addresses has a certain value, there is no such thing as a Bitcoin that can be copied. The combined value of addresses according to a Bitcoin block chain is a list of all transactions that have taken place on the Bitcoin network and it includes the order they have taken place. Every client could theoretically access this list.
A Bitcoin block chain is similar to a ledger listing every Bitcoin’s origin and present location. The block chain rules allow each Bitcoin to be present only once in a block chain, or else the block chain is not valid. Invalid ledgers are ignored by all Bitcoin users.
The only way to counterfeit Bitcoins would be to spend them in more than one place. This is called a double-spend attack. Because of the design of a Bitcoin block chain, (the list of all transactions) and the way the list is secured by mining, this would require a tremendous amount of computer power and is highly unlikely.
If a Bitcoin was copied, and shown to two users it would be obvious they are identical. There is only one public blockchain for transfers. The same Bitcoin may not appear twice except once as an output and once as an input.