Wireless Internet on a desktop PC

Hello,
my question is: if I want to configure wireless Internet on a desktop PC do I need both router and wireless adapter? Somebody told me that the adapter should be enough, so I've bought it, installed it and even if it works properly, it doesn't find any networks to connect to. I must underline the fact that I don't want to connect to a home network, but to a free external one. My office is located in an area provided with free wireless internet acces by the town's Council House. My colleague has a Laptop and he can acces the network, but I can't. Can you pls tell me what should I do? Thank you very much, I would really appreciate it.