A combi boiler, or combination boiler, simply put makes hot water and heats your radiators up in one unit. There is no need for an additional hot water cylinder with a cold fill cistern or a header tank for the central heating system. They can be identified easily by the amount of pipes, they have typically 6-7.
Also known as heat only, regular, conventional or floor standing boiler. They all require a separate hot water cylinder and cold fill cisterns (header tanks) to replenish the hot water supply and to initially fill and top up the central heating system. The hot water cylinder is normally located in an airing cupboard.
For homes with 2 or more baths. System boilers are pressurized with mains water and are accompanied with either a separate open vented or an un-vented cylinder. Rather than trying to heat a lot of water instantly they can be set to heat a larger volume of water indirectly. Vented cylinders have a tank above the cylinder in the airing cupboard or loft and rely on gravity to supply the hot water. Whereas un-vented cylinders are mains cold water pressure fed. The size of the tank is usually designed around how many bath and shower rooms the property has.
Still common to see, popular during the 1970's and 1980's when regular boilers were bigger and heavier. They were installed in a fireplace opening with a fire attached to the front. They are discontinued now. There was also a condensing version which due to the cost never really took off. A back boiler, like regular boilers, need a hot water cylinder and cold water fill tanks (header tank) to supply hot water and top up the heating system. Usually and easily converted to an combi, although converting to a regular is possible.