The mandolin is a small, teardrop-shaped instrument which actually evolved from the lute family in Italy in the seventeenth and eighteenth centuries. Although early Italian mandolins had 12 strings (six groups of two strings), the typical modern mandolin has four groups of two strings. Each set of strings is aligned closely together, so when fretting the instrument, your fingers hold down two strings at a time instead of one. The most common tuning for mandolins is GDAE, with both strings in each group being tuned identically. The GDAE tuning is similar to the tuning of a violin.
Most mandolin players hold the instrument by resting it on their slightly-elevated left knee (if they are right-handed). The right forearm holds the instrument against the body, leaving the left hand free to move about the neck of the mandolin. A pick is normally used when playing the mandolin.
One of the traditional ways a mandolin was played was through the use of tremolo picking. Because of the nature of the instrument, the mandolin cannot effectively sustain a note. To compensate for this, mandolin players will use alternate picking in to play a single note many times in rapid succession. For one of the most notable uses of this tremolo technique on a mandolin, listen to the Love Theme from the Godfather.
To get an idea of what the mandolin sounds like in a more contemporary context, listen to Rod Stewart's "Maggie May" (although they are present throughout the song, you can really hear the mandolin using the tremolo picking technique at about the 4-minute mark).
For more insight into the mandolin, see eHow's good tutorials on how to play mandolin.