Air rifles have a lead pellet that you load. Compressed air is then used to fire this pellet. This is done either by a spring in a chamber which you cock, and when you fire the spring shoots forward, compressing the air, which in turn fires the pellet. Alternately you can have a pre-charged air rifle, which had air a bit like in a divers tank. Here you just press the trigger, and these can be multi shot - mine has 7 pellets in a magazine.
Air rifles come in two sorts - ones any adult can own and ones which needs a firearms certificate (known as an FAC air-rifle). A non-fac air rifle can have air pressure up to 12lbs an inch, and will work an effective range up to say 20m if killing bunnies (more if you're good). An FAC air rifle can have much greater pressure, and much greater range.
A "normal" rifle has a bullet which consists of the bullet itself, and a cartridge behind which hold explosive. All rifles need a firearms certificate. Rifles come in all sorts of calibres (sizes). You simply load the bullet, and shoot. The most popular for bunnies is a .22mm rimfire, which equates well (ish) to a .22 FAC air-rifle.
Both air-rifles and normal rifles have advantages, as have shotguns, and if there was one answer to this question the others would not be made
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The decision is a mixture of cost, convenience, noise, and outright bias. Added to this will be a police assessment of your ground if you go down the FAC route (normal or enhanced air rifle).
For you, a shotgun is probably not the answer, lead around, probably neighbours to complain.
Personally I would consider a .22mm FAC air-rifle or normal rifle. However as above the police will consider both you and your ground to see if both are suitable. The key for the ground will be it’s topography – ie where you plan to shoot from, and where the bullet will go if you miss. If near neighbours, then police might consider it unsuitable.
You may do better to start with non-fac air rifle, and get some experience before progressing to more lethal means.