Have a Go!
There are a number of ways of making beer at home. Not surprisingly the slowest and hardest - full mash brewing - produces the best beer! There is some consolation though in that it's also the cheapest method, especially if you're a keen DIY-er. Most people start with a kit though. As you progress to full mashing all of the equipment you acquire is still useful though, so you can start with a kit and won't need to discard anything.
Beer Kits
There's nothing simpler than picking up a 'beer kit' in a hardware shop, mixing it up in a bucket and producing some mediocre beer! Disheartening? We don't want to get your expectations up - even producing something of the quality of John Smith's Extra Smooth will require a top-quality kit in good condition, excellent sanitation and for heavens' sake don't put it in the airing cupboard! If it doesn't go, add more good quality yeast.
Help is at hand - this article at Jim's Beer Kit will get you off to the best possible start - good luck!Extract Brewing
If you've been searching the internet you will have seen a lot about 'extract brewing'. Like a kit, you buy a couple of tins of malt extract, but you make a beer more to your taste by boiling it with hops. This provides a more stable and tastier beer. While popular in the US, it is not a particularly common method of brewing in the UK. However if you want to brew a better beer than is possible from a kit it's a step forward - the only extra equipment you'll need is a big stockpot - 20 litres or more - or a home brewing boiler.
Have a look at this article at Jim's Beer Kit for more information.
Mashing
The full mash brewer takes malted grain, steeps it in warm water and then drains it off to produce a malt 'wort' - a lovely smelling sticky substance. They then boil it with hops to add bitterness and aroma, cool it and ferment it with yeast to make beer. Sounds familiar? It's exactly the same process that the world's best brewers use, just on a smaller scale.
Again, there are good instructions to get you started in Jim's Beer Kit.
In US-influenced circles, full mash brewing is known as 'all grain' brewing.