I have 1 laptop, 1 wireless router, and I want to make the laptop wireless. Do I need another computer to do this?

Many of the newer laptops have a button at the top, near the power button that has a little "antena" symbol, most the time all you have to do is press that, and the computer will find the wireless service all by itself.

Wireless button can be on the side as well. If it doesn't have wireless built in, you could use a USB adapter.
This is a little big, I like Linksys, should work well:
http://cgi.ebay.com/Linksys-WUSB300N-Wireless-N-WiFi-USB-PC-Adapter_W0QQitemZ140312263838QQcmdZViewItemQQptZLH _DefaultDomain_0?hash=item20ab433c9e&_trksid=p3286.c0.m14&_trkparms=65%3A12%7C66%3A2%7C39%3A1%7C72%3A1234%7C 240%3A1308%7C301%3A1%7C293%3A1%7C294%3A50
Here is a smaller one, both can be plugged in directly or with a stand:
http://cgi.ebay.com/LINKSYS-WUSB54GSC-Wireless-G-USB-Adapter-w-SpeedBooster_W0QQitemZ220427150009QQcmdZViewItemQQ ptZLH_DefaultDomain_0?hash=item33527b62b9&_trksid=p3286.c0.m14&_trkparms=65%3A12%7C66%3A2%7C39%3A1%7C72%3A1234%7C 240%3A1307%7C301%3A1%7C293%3A1%7C294%3A50