Things to take into consideration:
Price: Most desktops and laptops based on gaming run high in price. While a laptop will be more expensive in the long-run, they're also not very suitable for running the best of games very well. A desktop gives you the freedom to upgrade, while a laptop does not considering disassembly and graphics cards being soldered to the motherboard. To lower the price of a desktop, you can buy all of the pieces and assemble yourself, though without knowing what you're doing, you're 75% sure to break something. Sometimes building one may be cheaper, but you'll spend more in the long run on parts you'll have to replace because of your own misdeed. No you can't return them! For a gaming laptop look to spend 700-1500 as a median price. For a desktop, 500-1000 should suffice for mid needs. If you plan on upgrading do NOT buy a mini, or half tower kit that HP or Emachines releases.
Capabilities: How well do you want this thing to run certain things? For minecraft.. you can do a 400 dollar laptop easy and be good to go. If you're looking to play higher end games such as a few titles sold on steam, you'll need to get a desktop. A laptop WILL ABSOLUTELY NOT have the proper ventilation and you run the risk of destroying the system from overheating. I have a core i7 quad laptop with an Nvidia 330m and still run into lag and slow loading even on TF2. For gaming I seriously urge you to go the route of a desktop.
Need: You should never buy a computer based on gaming alone. What else will you need it for? Are you an artist? Do you go to school? do you want to do something that may warrant using a high CPU program? If the question is something you have to think about, go the route of the desktop. At least then you won't have to buy a whole new system
On those points, who's to say you can't get a desktop and potentially get a cheap laptop later? You can have your cake and eat it too when it comes to computers.
Edit: this topic keeps popping up. I'm gonna make a huge thread about this so that people can think on it without a bunch of new threads popping up.