Confirmation:
After a transaction is broadcast to the Bitcoin network, it may be included in a block that is published to the network. When that happens, it is said that the transaction has been mined at a depth of one block. With each subsequent block that is found, the number of blocks deep is increased by one. To be secure against double spending, a transaction should not be considered as confirmed until it is a certain number of blocks deep.

Number of Confirmations:
The classic bitcoin client will show a transaction as "n/unconfirmed" until the transaction is 6 blocks deep. Merchants and exchanges who accept Bitcoins as payment can and should set their threshold as to how many blocks are required until funds are considered confirmed. Most trading platforms that bear the risk from double spending require 6 or more blocks.