Western Union sucks.
At the "outlet" as Judy calls them, they will have paper versions of the western union form you have to fill out. On the form you'll need the person's name, address (of where they will be picking up the money from), how much you are sending them, etc. After you fill it out, you'll need to pay (most likely in cash) the amount that you'll want to send, plus the fee that they'll charge to send it.Then they'll give you some sort of code that you'll need to give (whoever is picking up the money), so they can tell western union on their end what the code is, so they can pick it up. Usually it takes a few hours or so before the cash will be available for them to pick up at their location.
Sending through Paypal is cheaper. If there is the same bank in their country as where you are, you can just deposit money into their bank account (no id required to deposit). Many banks have bank transfer things you can do online.
Judy is correct. Many supermarkets and discount stores do Western Union–generally at their service counter.
Before sending any money via Western Union, be absolutely sure that you're not part of a scam. Once you send the money and it's picked up by someone, there's no way to get the money back. If you're sending the money to someone you met over the Internet or responding to some sort of Internet scheme, you'll be losing the money. Generally only consider sending money to a family member or to a friend you met in person, and not just over the Internet. Remember there are no US soldiers stationed in Africa who need money to get home, and there are no African bank accounts that you could get hold of by sending money to someone who will split the money with you–those are all scams.
Take cash to one of their outlets. Many large grocery stores have them.