The cryptography algorithm bitcoin uses is known as asymmetric cryptography. In this algorithm we have a "key pair" which is a duo of a key that should be kept privately (called private key) and another key corresponding with the first one that can be revealed to the world and shared publicly (called public key).

In bitcoin when you want to pay someone you have to get their public key then create some sort of "lock box" where you put the coins in that only their private key can unlock (provide a valid signature).
To make things easier and to hide the public key, we use the hash of it and then encode that hash with a human readable encoding. The final result is the address.

In short you need the address to make or receive a payment.