As others have said "copying a bitcoin" is trivial but of no value.
Your wallet consists of addresses and each of those addresses has a certain value.
When your wallet says you have "20 bitcoins" it simply an abstraction. Currently your wallet file consists of multiple addresses and the combined value of those addresses according to the bitcoin block chain is 20 bitcoins.
When you transfer bitcoins to someone you don't hand over "a bitcoin". You submit a transaction to the network. The network makes sure your address is valid and has the proper value. So there is no risk of counterfeiting because there is nothing to counterfeit.
There is a risk of double spending. When I transfer bitcoins from me to you the network prevents me from doing it again (and again and again ...). When you submit a a transaction the network verifies that each transaction is valid before including it in the next block.
IF hypothetically you had enough computing power you could cheat. You could include your invalid transaction in the block and sign it as valid. The strength of bitcoin comes from the size of the network. Currently ~17Thash/second. An insane amount of computation power. As long as the network remains large enough and diverse enough that one entity can't control 51% there is very little risk of double spending, so long as the recipient waits for a confirmation.