A key difference between traditional electronic currency and Monero is that your account resides only under your control, normally on your computer, and cannot be accessed by anyone else if you [practice good security](#practicing-good-security).
Your account contains all of the Monero @transactions you have sent and received. Your account balance is a sum of all the Monero you've received, less the Monero you've sent. When using Monero you may notice that your account has two balances, a locked and an unlocked balance. The unlocked balance contains funds that can be spent immediately, and the locked balance contains funds that you can't spend right now. You may receive a transaction that has an @unlock-time set, or you may have sent some Monero and are waiting for the @change to come back to your wallet, both situations that could lead to those funds being locked for a time.
Your account is based on two keys, a @spend-key and a @view-key. The @spend-key is special in that it is the single key required to spend your Monero funds, whereas the @view-key allows you to reveal your @transactions to a third party, for example for auditing or accounting purposes. These keys in your account also play an important role in Monero's @transaction's privacy.
The private keys for both of these must be protected by you in order to retain your account privacy. On the other hand, the public keys are obviously public (they are part of your Monero account address). For normal public/private key cryptography someone could send you a private message by encrypting it with either of your public keys, and you would then be the only one able to decrypt it with your private keys.