When you use the bitcoin protocol to transfer value to someone you don't know, you need to make sure that you have something to communicate with that person to transfer the payment. This is called a bitcoin address. You may be aware that it is the same thing as throwing a rock from a bridge into the water. It can't go anywhere else.
The example that comes to my mind is this one: if you buy a product on Ebay after winning a bid, you need to indicate where to send your package after paying the buyer. The critical information that you need to give is a valid address, otherwise, you will never receive what you have paid for.
In the bitcoin ecosystem, we call this a bitcoin address. Don't worry, your wallet will generate it for you. You don't have to memorize it otherwise, you will need a bottle of aspirin. This address is composed of 26 to 35 alphanumeric characters. As you can see, you need to be a super brain to memorize a string of characters like this. Myself I hate to learn stuff like this by heart.
You can check on the main page of this website https://blockchain.info/ for examples of bitcoin addresses. You can see an example of the string of characters that I mentioned earlier. If you go to the website you have the opportunity to click on one of those address and have the complete details of transactions occurring in the blockchain.
Here is one of the transactions that are occurring into the bitcoin blockchain. You have the address on the left that starts with the number 3. Normally those addresses start with the number 1 and 3, and the rest of a complete string of characters.
The bitcoin address is the public key that your wallet client will generate for you. In another word, the keys will come from the asymmetric key pair. You have two keys in your possession, the private key that you need to keep in a safe place and you use to sign the transaction with it. The public key is the one you provide for receiving the payments from the applications you use. You need to have a good match between the public and private key for the transaction to be complete. Without an address, the bitcoin ecosystem can't work and nobody can send transactions over the network. You need to keep in mind that an address is very basic and essential for the bitcoin protocol to work.
Even for myself, it is a hard concept to understand, the area of cryptography is something scary for me ;). So if you want to send bitcoins to a friend or make a payment, you need to have the public key from the other party involved in the transaction.
Hope you have a better comprehension of what a bitcoin address is. If you want more information, I will be honored to answer you in the Disqus comment section below.