Each payment carried out on the Bitcoin network is identified through a set of two keys, randomly derived. The private key can be compared to the password of an account: it is used to prove the ownership of the associated address and thus to fulfil the necessary conditions to spend the funds in question. It gives a user full control over his funds and is never to be shared with anyone. Having a backup of the private key is fundamental to recover the funds in case of unexpected events (e.g. hardware failure).
The public key is instead to be compared to an e-mail address or a user name. Unlike the private key, it can be shared. It is also used to derive all the Bitcoin addresses where a user receives transactions.