Compaq CQ60 Laptop Won't Connect To The Internet Wirelessly

I have a Compaq CQ60. It formerly belonged to a friend (who had no problems with it) and I restored factory settings. However it isn't connecting to my router! Well, it does connect - but it calls it "unidentified" and despite my many attempts it just won't allow me access to the internet. The two other laptops in the house are connected to it fine though. I tried connecting to my neighbour's wireless router, and it worked no problem, including the internet! I bought an ethernet cable earlier, which successfully connected my laptop to the internet. (Although it said I was connected to a homenetwork as well as my Talk Talk router, which it proceeds to call "unidentified" and is restricted to local access). However, I really need it to work wirelessly. I have not been able to use the laptop yet as the ethernet cable does not allow me to leave the router!