What is a Mandolin?
A mandolin is a fretted instrument with eight strings grouped into four courses, each tuned a fifth apart.. In each course both strings are tuned to the same note, but the impossibility of them being totally identical provides an interesting chorus effect, which helps to make this tiny instrument sound so big. These courses are tuned (low to high) G-D-A-E, the same tuning as a violin. If you can play a violin, and have a basic grip on how a guitar or ukulele should be played (and don't have huge fingers) you can play a mandolin. If you can play guitar or ukulele, and have a decent ability to think your way around different tunings, you can play mandolin.
Most mandolins resemble a miniature acoustic guitar, only with a differently shaped body and a neck profile a little more like a violin (narrower, and with more fretboard camber). An average mandolin is approximately one and a half feet long and about ten inches across. The standard scale length of a Gibson mandolin (fairly strong representative of the bulk of mandolins around) is 14 1/8".
How Do You Play a Mandolin?
I'm not going to write a comprehensive guide since I'm an abysmal mandolin player myself, but the general theme to mandolin playing is quite similar to other fretted instruments. Hold the string against the fret and pluck the string. Do this with more than one string at once to get a chord.
Mandolins don't have much sustain, so rather than plucking and holding a note like you would on a guitar, tremolo picking (fast back and forth picking on a single note - think of The Godfather soundtrack and you're right on the money) and fast strumming tend to keep the level of sound coming out of the instrument right up there.
Types of Modern Mandolins
There are two common styles of mandolin to be found in modern music, the A style and the F Style:
The A style (designed by Orville Gibson) is the (slightly) older and (significantly) cheaper design of the two, and has a more or less teardrop shaped body, with an almost totally flat back, an arched top, a trapeze style bridge, and two f-holes.
The F style (ALSO designed by Mr Gibson) is very similar to the A style in terms of functional design, but features a short, sharp cutaway horn on the treble side of the body, pointing at about 45 degrees away from the neck, and a large intricate curl on the bass side of the body. F styles usually have only one f-hole, but are usually more beautifully lacquered and bound, sometimes with intricate inlay work in the headstock and fretboard.
Before Gibson's creation of the modern mandolin at the beginning of the 1800's, most mandolins were bouzouki-style affairs with large rounded backs, but were not otherwise very different from their more modern counterparts.
Some companies (Fender for example) also make solid body electric mandolins, some with variations in the number of strings/courses (just to make things hard for us all).
The mandolin family also includes the mandola (same deal as a mandolin only larger, and tuned down a fifth overall), the octave mandolin (larger still, only an octave below a conventional mandolin), and the mandocello (even larger, and tuned an octave below a mandola). The observant amongst you might have noticed that this lot roughly corresponds to the violin, the viola, the viol, and the cello. I've never heard of anyone with a bass mandolin, but I have no doubt that John Paul Jones of Led Zeppelin fame has commissioned one to be made for him. He's had bass versions of everything else made, so why not a mandolin?
Where to Hear Mandolins in Use
Mandolins feature most heavily in a modern setting in bluegrass and country/blues music, but (like their cousins the guitars) have been used in a huge number of different styles of music.
In terms of rock bands, Led Zeppelin's 'Battle of Evermore' features some beautiful cascading mandolin sounds, and Cheap Trick's guitarist Rick Nielsen made regular use of solid body mandocellos, in one case as one neck on a custom quintuple-neck guitar made for him by Hamer Guitars.
In country/bluegrass music, slugo cites Bill Monroe and Ricky Skaggs as examples of players who "can really demonstrate how to tear one up."
The most prominent use of mandolin in "classical" music (note the quote marks) was probably baroque composer Antonio Vivaldi.