Two factor authentication is a way of verifying your identity before allowing you to access your account. It requires you to verify your account through two different mediums before allowing you to log in. Whereas the first medium usually requested is a password, the secondary medium usually asks for the input of some value only known to you - it asks for information that others would not able to look up in an attempt to impersonate you.
When implemented, it adds an extra level of security to your account. Think of it like a transaction at an ATM; when you insert your card into the machine, the card alone does not grant you access to your funds. Usually, ATMs require that you also supply you 4-digit pin code in order to identify that the cardholder is the same as the account holder. The account is opened to the user only after positively affirming the identity of the user through the two different mediums.