yes you need a wireless router, but what kind you need depends on what internet connection you have - cable or ADSL.
If you have a cable modem you are in a better situation as the cable modem will plug into the router, but if you have ADSL you are probably using a USB modem, which won't be of any use, and therefore you will have to buy a wireless router with built-in ADSL modem.
I'm not sure if there prices are necessarily keenest, but www.broadbandbuyer.co.uk/ have been the best site to look at this stuff for a long time.
I've been through quite a few wireless routers, they are considered 'consumer grade' technology, which means that the manufacturers don't bother testing them properly before releasing them. What this can be mean is that you lose connection, and have to unplug them all the time.
It sounds like you will have the router next to the desktop, and so that will be connected using a cable (not wireless), and only the notebook will connect wirelessly.
The biggest issue with wireless routers is that they suffer badly with speed and signal drop-off with distance. The latest wireless standard, called 802.11n (or just N), following on from 802.11g, and the basically obsolete 802.11b, has improved range (useful if you want to use your laptop in the garden, or upstairs through thick walls).
If you buy a 802.11n router, you will need an 802.11n laptop to make use of the improved speed (802.11b or 802.11g devices can still work, they just won't get any benefit compared with an 802.11n router). Centrino-branded laptops sold since late 2007 will support 802.11n. Older/non-centrino laptops might not.
As for which one to buy, I had a Belkin which was rubbish, don't rate Netgear, but my Buffalo is great, I have a more expensive version of this:
uk.insight.com/apps/productpresentation/index.php?alert=categoryresults&product_id=BUFNJRHPGN
which is a bargain TBH.
If you have cable rather than ADSL, get this one www.broadbandbuyer.co.uk/Shop/ShopDetail.asp?ProductID=7561