You can expect Bitcoin payment notification in seconds after your customer clicks "Send" button in his Bitcoin client. While technically the transaction is not confirmed by the Bitcoin network, for most purposes it is practical to consider it so. Support for "confirmed" transactions is being developed for those cases when you need to be 100% sure (and do accept delays).
Do you see it now?
Considering how many people trusted mybitcoin, and a startling willingness to abandon a key security feature of bitcoin I think it's important that everyone understand that security is a multifaceted problem.
are now fully supported in
BitcoinNotify.
Merchant decides how many confirmations are "enough" for his/her purpose.
It is also possible to order notifications on
several confirmation levels, i.e. 0/unconfirmed for visual payment acceptance in the store, and 6 confirmations for actually shipping the item.
PS @MrJoshua: thanks for stressing the importance of security in this thread.