You’ll almost certainly have a heat pump in your home already, as they are used to keep fridges cool - basically by ‘pumping’ heat out. Most air conditioning units are also heat pumps. Both of these are generally airsource systems.
A ground source heat pump will usually be the more efficient option for home heating – giving the lowest running costs. Air-source systems are cheaper to install, but their efficiency will drop as air temperatures drop (when you need heat most). Also, heat transfer from air is more difficult than from other sources. Water-source systems can be very efficient, but they’re not common because you need a water source that will not freeze (such as a spring).