Choosing a computer used to be more simple than it is today. You would look at the speed of the processor (CPU), the amount of memory (RAM), the size of the hard disk drive and possibly the type of graphics capability. They would give you enought information to decide if the computer would suit your needs. Modern computers, however, bring more factors into play. For most people, the key factor is the available budget - the more powerful a computer, the more it is likely to cost.
Perhaps the first point to consider is whether a desktop or notebook (laptop) computer would be more suitable. We offered some thoughts on this in a previous post "Why would anyone choose a desktop PC?". For the purposes of this discussion, it is worth mentioning that mobile and desktop CPUs are not the same. The mobile processor will use various methods to minimise the amount of power it uses, so mobile processors tend to be slower than desktop equivalents.
The performance of the CPU depends on a number of factors: the speed of the processor core(s) (often known as the clock speed), how fast it can communicate with the rest of the computer system (via a HyperTransport or QuickPath Interconnect link), the size and organisation of the processor cache (high-speed memory in the CPU itself), the number of processor cores (most modern processor packages have more than one processor unit) and whether, in the case of Intel processors, HyperThreading is included (which allows one processor core to do two things at once some of the time). There are, of course, other factors, but these are the main ones. In general, bigger numbers are better (higher clock speed, faster interconnect, more cache) but more expensive.
The next factor to consider is the hard drive. As they are mechanical devices, hard drives haven't increased in performance at the same rate as the electronics in the computer, so the hard drive can be a serious performance bottleneck. Many notebook computers use drives spinning at 5,400r.p.m. wheras desktop computers usually have drives spinning at 7,200r.p.m. making them significantly faster. To further complicate the picture, Solid State Drives (SSDs) are available on higher-end machines. While spinning hard drives are mechanical, SSDs are electronic, and much faster, though also much more expensive (although prices will fall over time).
Assuming your new computer will be using Windows 7 or Windows 8, you should look for a machine with a minimum of 2GBytes of RAM, and more if possible. If you have less, the computer is likely to be very slow, even for basic tasks. For basic office and web browsing tasks, 2GBytes should be sufficient. If you want to process video or play games, you should be looking for 4GBytes or more.
The size of the hard drive is not an issue for most users. A 250GByte drive (the smallest you are likely to find) can store over 50,000 pictures! However, if you have a large music collection, or you want to install a lot of games, you will probably need 500GBytes or more. If you want to store video, you will run out of space no matter how big your hard drive is!
For gamers, the other factor is the graphics card. This area changes so rapidly that the best advice is to do your research and buy the best you can afford. Bear in mind that if you want to use multiple cards, you need to check that the computer you are considering will support your preferred setup.
As a final (non-technical) point, consider how the computer will fit into your home. If it is going to be in a family area, the way it looks could be an important factor for you. Some computers are designed to fit in under the TV or in a corner of the living room. This is a purely personal choice that you'll have to work out for yourself.