If you know what the parts of a computer are, and what they do, it makes it easier to choose something that suits your needs.  Below is a list of the more important things.

Processor – The engine of your computer.
Intel Atom  - For basic users, Windows Xp and Net Book Computers.
Intel Celeron Dual  - Basic and home users.
Intel Pentium Dual  - Home users and medium speed.
AMD Athlon x 2 -  Home users and medium speed.
Intel Core2 Duo -  Home users, Faster speed .
Intel Core2 Quad -  Enthusiasts , fastest.
AMD Phenom Quad  - Enthusiasts, fastest.

Ram/Memory - is the place in a computer where running program data is kept so that it can be quickly reached by the computer's processor.  If you do not have enough ram, your computer will get slower the more you try to make it do.
Ram comes in many different sizes, however most users with Windows Vista will need between 2-4 gigabytes.
Windows Xp runs fine with between 512 Megabytes and 1 Gigabyte of ram.

Hard Drive – Is your computers main storage.  This is where everything on your computer is saved.
These days hard drives come in sizes between 160 Gigabytes to 1 Terabyte. 
Most home users will find 160 Gigabytes sufficient, however if you intend to store huge amounts of data, then the bigger the better.

Graphics Card - If you wish to play high quality games, it is important to have an Nvidia or Ati graphics card with a dedicated memory.

Anti Virus - It is very important  to protect your new computer with Internet Security software such as, Kaspersky, Norton or Bullguard.  Make sure what you buy has a firewall built in.  Antivirus is something you cannot go without.

Office - No computers come with MS Office installed these days, so it is worth considering purchasing a copy of this if needed.  The basic version of office called "Home and Student" includes Word, Excel, Powerpoint and Onenote and costs around $230.

Backup - You should also invest in either a usb pen drive or portable hard drive, to backup your important data, such as photos and documents.  If you don't do this and your computer gets a virus, or hard drive fails, you may lose it all.

Insure - An extended warranty is also recommended as computer repairs can be costly, but make sure the price is proportionate to the computer.  For example, no more than quarter the total computer price.

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