Always read the listing what the computer can do and if it can be upgraded later on...
If it's for gaming an really good CPU like 4x 2.8GHz or better with a PCI-E slot for a really good graphics card in the mother board and can go up to 4GB of Ram and large hard drive like a 160GB or bigger for storing those games on.
If it's for a office an 2x CPU like a 2.6GHz with a large hard drive like 500GB or more and good modem or broadband modem connection for the internet.
For the home computer any of the above will do or a older computer like a P3 866MHz will still do the job for photos or printing documents or writing a story book.
Choosing a computer for what you want can be a bit complex from time to time, but when you get the right computer you will be happy with it.