There are many different types of bagpipes, including several from the Celtic nations. Here's a brief description of some of the more common ones, particularly those from the British Isles.
Scottish smallpipes (SSP)
Bellows-blown (though occasionally mouth-blown) pipes, with drones from a common stock and a parallel (straight/cylindrical) bore chanter. They are usually in the keys of A, D, C, Bb & sometimes G.
Pastoral / Union / Uilleann Pipes (UP)
Bellows-blown conical bore chanter extended with a foot joint, and able to over-blow giving almost two octaves. The three drones from a common stock are each pitched an octave apart, and there are none or more regulators (keyed drones). They are usually in the keys of D or C.
Northumbrian smallpipes (NSP)
Whilst technically these pipes are not Celtic (i.e. not from one of the Celtic nations), they are a close cousin to some of the varieties of Celtic pipes, so we include them here too.
“Bellows-blown smallpipes with a closed-ended parallel bore chanter. A common set consists of bellows, bag with four drones, tuned D, G, d, g, and a 7-keyed chanter. The drones may have tuning beads, commonly one on each drone, so that they can be retuned to E, A, e, a. Only 2 or 3 drones are used at a time, but this extends the range of possible keys from 2 (G and D) to 4 (D, E minor, G, A minor). The chanter has a range of almost 2 octaves (not chromatic) and it is possible to play a wide range of material on it. They are usually in the keys of F+ (20 cents sharper than F), F# & G. Most keys are available.
Lowland / Border pipes (BP)
Bellows-blown pipes, with drones from a common stock, and conical (or tapered) bore chanter. (Highland Reel Pipes and Northumbrian Half-long Pipes are essentially the same). They are usually in the keys of A, G or Bb.
Shuttle pipes (ShP)
Smallpipes, usually bellows-blown, with a double reed drone arrangement drilled into a single solid block, and each drone tuned with a slide. They are usually in the keys of A or Bb.
The term ‘Lowland Pipes’ refers to all Scottish bellows-blown and mouth-blown pipes with drones from a common stock, including BP, SSP and Pastoral Pipes, but excluding Highland Reel Pipes. (Sometimes LP are defined as only those pipes with a conical bore chanter and drones from a common stock, i.e. BP).
Link to Pete Stewart's 'Life and times of the Lowland pipes' article published in the Piping Times.
Cauld wind pipes
This is a general term sometimes used for Scottish bellows-blown (‘cold’ wind) pipes.