Computer games are probably the most intensive programs used in peoples' homes, but with the proliferation of consoles, it's not always obvious that the computer spec is important. People can assume any computer will place games as this is after all what consoles do, but this is not the case.
The important parts
All typical computers will consist of a large box into which are plugged your monitor, printer, keyboard, mouse, internet connection, speakers and any gaming controllers. When buying computers and laptops, it is the main box, not the peripherals that plug in that matters. Inside your box or laptop you generally have the following components: CPU The CPU or processor does much of the hard work inside your computer. These days, most processors can cope with games playing but two model ranges are definitely the best choices. GPU The GPU, Video or Graphics card is the heart of displaying the game on your screen. This card is fundamental to the performance of your computer because graphics are hugely complex in most modern games. RAM RAM (Random Access Memory) is the temporary storage used by your computer. Typically 2 or 4gb (gigabytes) will be more than enough for most games. HDD HDD (Hard Disk Drive) stores your games, game saves and other data. While they can have a performance hit on the loading time for games, overall the big thing to consider is the capacity. It's probably best to look at a minimum of 200gb storage. Other components are fairly standard and include the DVD drive, motherboard, network card and sound card. The sound card and network card are both normally integrated into the motherboard. Sound cards can be improved but for gaming, the integrated ones are normally fine.
What to look for in the important bits
As you can see, the really important parts are the CPU and video cards. Two main manufacturers supply CPU's, Intel and AMD and both have pros and cons. AMD are cheaper and provide CPU's with very good performance but not quite as good as the more expensive Intel alternatives. Most of their CPU's will be fine for gaming but it's best to avoid Celeron, Sempron and 'Pentium' processors as these are usually 'cut down' or older technology devices. Phenom and Core i5 and i7 are excellent choices AMD/ATI and Nvidia are the two choices and they can be a bit of a minefield. The best thing to do is to look on Tomshardware.com at their 'Hierarchy' guide. Any card in the top half of the guide are good choices, although free screen gaming will need cards at the top of the guide. Research is the key.