Split vs. monoblock
Heat pumps can either be 'split units' or 'monoblock' units. As the name suggests, split units consist of two separate components: an outdoor unit that collects heat from outdoor air, and an indoor unit that transfers the heat into your radiators. The two units are connected by refrigerant piping which carries the heat from the outdoor unit to the indoor unit. The outdoor unit consists of a heat exchanger (to extract heat from the air) and a fan (to push air over the heat exchanger so that it can extract as much heat as possible). The indoor unit is a similar size to a gas boiler and consists of a second heat exchanger where the hot refrigerant transfers heat to the water in your heating system. Monoblock units have this second heat exchanger within the outdoor unit, so you do not need an indoor unit. In this case hot water is just piped directly from the outdoor unit.
Air-to-water vs. air-to-air
Air source heat pumps can either be 'air-to-water' or 'air-to-air'. The first word describes where the systems get their heat from, so will always be air for an air source heat pump. The second word describes the substance that the heat pump is heating up. In a typical British home, our heat is delivered by water flowing through radiators, and so our heat pumps need to heat up water. They are therefore air-to-water heat pumps. In this case the heat pump heats the water, which circulates through the radiators and heats the air in the room. An alternative is to heat the air directly - either with in-room 'fan coil' units that look like hotel room A/C units, or by carrying air around in ducts (like the things people crawl around in spy movies). Air-to-air units are common in commercial buildings, and in homes in countries where air-conditioning is widespread like the U.S. In the UK they can be a good choice for heating isolated rooms, or for homes that are likely to need cooling, such as flats or homes with loads of windows. They are typically cheaper and more efficient than air-to-water heat pumps (because they don't need to make the heat as hot), but they do require a suitable distribution system and they won't heat your hot water, so you'll need a separate heat pump water heater for that. Air-to-air units do not meet the criteria of the Boiler Upgrade Scheme grant, however fan coils supplied by an air to water unit do meet the criteria.