If you want to game and would like to get good bang for your buck get a Desktop:
- Way more power at the beginning
- Fairly easy/cheap to upgrade
- If you have a problem with a part, you can replace it easily (by yourself), with a laptop you are usually screwed
- Quieter (usually because the PC is under the desk, the laptop a foot away. Also water cooling (if you have one) is a lot quieter as well
- Keyboard is cool, gaming laptops will often inadvertently heat the keyboard
Laptop if the above doesn't bother you and you are prepared to buy a new laptop once yours is outdated, rather than upgrade it. A gaming laptop is NOT very mobile. Low battery life time (even if not playing games), heavy and big (for a laptop )
If you need a laptop anyway for lectures or so, also don't get a gaming computer, the power consumption even in idle mode makes them very often unsuitable for this. Get a netbook instead, they last 6-10 hours and are fairly cheap.
Also on a desktop a dual screen setup is more comfortable and very worthwhile (and cheap) to get. I have two 21 inch, bought them for 150 pounds 5 years ago.