By far the most popular type of boiler in British homes is the compact combi boiler, and with good reason. They are small and economical, and provide an instant, endless supply of fresh hot water without the need for heating a tank.
The combi boiler is a good solution for a small house or flat with only one bathroom. Also, if you have a big cold water tank in the loft that you would like to take out in order to create more space, then a combination boiler could be a good way to go.
A system boiler, also known as a closed or sealed system boiler, stores hot water in a tank in an airing cupboard. It is a good choice for homes with multiple bathrooms, as you can run several hot taps from it at once, and it has the advantage that it does not require a separate tank in the loft.
In the past, the open vent boiler has been the most popular central heating system in the UK. It requires two tanks in the loft and one in the airing cupboard, so it is very space-intensive. Open vent systems allow any number of hot water sources to be drawn at any time, and are a good choice for large houses with a number of bathrooms and sinks. Water pressure can be a little low though, and the hot water can still run out if you use it too quickly.
It is not recommended that you buy a back boiler unless it is a replacement for a similar unit, modern back boilers are a lot more efficient than older ones. It could save you a lot of expensive plumbing work, not to mention the money you would save on fuel, if you do replace your old back boiler with a more modern equivalent.