Condensing boilers

Nearly all new boilers are now 'condensing' types.  Condensing boilers are far more energy-efficient than traditional boilers, as they reuse heat that would otherwise be wasted.

Some heat generated by traditional boilers disappears up the flue in the form of hot waste gases. Condensing boilers use some of the heat from these flue gases to heat water, making the boiler far more efficient.

Combination boilers

These provide heat for your radiators and domestic hot water on demand.  Hot tap water comes directly from the mains and is heated as it's used, so there's no need for a separate hot water storage cylinder or cold water feed cistern. However, there is a small time lag between turning on the tap and getting hot water, and the flow rate is limited by how quickly the boiler is able to heat up the water.

The advantage is that they require less space than heat-only boilers (because there is no storage cylinder), this avoids heating water that is not used immediately. On the down side, the flow rate limits how much water you can use at one time; e.g. no filling the kitchen sink while running a bath.

You could also miss using the (wasted) heat from your hot water cylinder in your airing cupboard.

Conventional heat-only boilers

Heat-only boilers provide heat for radiators directly, but they need to be connected to a separate hot water storage system including a hot water cylinder, cold water cistern and in some cases an additional expansion cistern.

These arrangements take up more space than a combi system, but are able to provide hot water to several sources at once and are more suitable for larger houses.

Unless a cylinder is very well insulated it usually loses heat into its surroundings, thereby wasting energy, and the amount of hot water you can use is limited by the size of your cylinder.

For instance you may run a bath, then need to wait for the boiler to heat up the water in the cylinder again before you can run another.