This is the first question you'll need to ask yourself. Think about where you're going to be riding regularly and also consider any future possibilities. Are you going to be on the road only, would you like the option of being able to ride off-road, or maybe it would be nice to be able to go for a ride along the well used paths on the Downs in the summer.
You'll also need to think about whether you'd like a bike that will get you from A to B as quickly as possible, or maybe something that will be a little more comfortable at the sacrifice of speed.
Your budget will obviously be a big deciding factor, but as a general rule £300 is about the least you'll need to spend to get a mechanically reliable bike. Above that the bike become lighter and the components more responsive giving the bike an overall more responsive feel. The frame materials will become more advanced as you pay more money, but you will only get what you pay for, so don't be fooled by some brands that appear to be too good for the money. There are no benefits to be had from cheap carbon frames, or cheap suspension bikes, in fact, there are usually more problems. Expect to pay over £1500 for a good quality and well manufactured carbon framed bike and over £1000 for a full suspension bike that will perform well.
The most important advice is, RIDE IT BEFORE YOU BUY IT! Try as much as you can within your budget, you'll notice the differences and soon pick out a favourite.
A road bike is like the traditional racing bike, like that you see in the Tour de France. It will have a dropped handlebar, a very light frame, skinny wheels and tyres. These bike are designed to be used on smooth road surfaces only. There are different types of road bikes with slightly different geometries for different riding experiences. Most common are those designed to be the fastest and those which are slightly less fast, but designed to be very comfortable for all day rides.
The commute bike is designed to be fast and comfortable on the road. It has similar wheels to a road bike, but a flat bar like a mountain bike and a riding position that is more upright. Although not as fast as a road bike, they are light and nippy enough to get you around in good time. They also have fitting points for mudguards and racks that some of the more expensive road bike won't have.
City or hybrid bikes as the can be known are designed to be ridden predominately on smooth road surfaces but will usually be capable of being ridden on dry unmade paths, canals tow paths, ant the smoothest bridleways. They use the same wheel diameter as a road bike but a much fatter tyre. The tyres can be smooth or a little knobbly if they are likely to be used on loose surfaces. These bike have a fairly upright comfortable riding position, making it easy to maneuver in town and better for less experienced riders.
A hardtail mountain bike has suspension at the front, but a rigid rear end, like any conventional bike. The mountain bike has for a long time been a popular choice as it can be ridden almost anywhere. If you want to head off road and have some fun, a mountain bike is essential. The down side is it will be less efficient to ride on the road, worth bearing in mind if you're going to be riding it to and from work everyday of the week. It is possible to put a wide range of tyres on to a mountain bike from skinny slicks up to big knobbly tyres. The riding position is designed to give the rider good control of the bike over rough terrain. As you pay more money, the riding position becomes more 'aggressive' putting the rider in a position to pedal more efficiently, making the bike feel more responsive.
Full suspension mountain bikes have suspension front and rear. There are many different types for different styles of off-road riding. There are shorter travel (l00mm or less) bikes that are normally aimed at racers wanting the best pedaling efficiency and the fastest ride. Mid travel bikes (100 - 140mm) are usually designed for general cross-country use and not quite as efficient or fast as the shorter travel bikes. Next up are what are generally known as all-mountain bikes (140 - 160mm). They have more travel and a more 'relaxed' geometry that suits downhill riding, but they are still able to be ridden relatively easily uphill. Then there are downhill bikes (160mm +). These beasts are usually heavy bikes designed to be ridden downhill only as fast as possible, not a great choice if its your only bike!
Folding bikes do just that! Most common and useful are the type that will fold down to a size that will enable you to carry them onto public transport.