What is a multiwire branch circuit?
The National Electrical Code defines a Multi-wire branch circuit as follows.

Branch Circuit, Multiwire. A branch circuit that consists of two or more ungrounded conductors that have a voltage between them, and a grounded conductor that has equal voltage between it and each ungrounded conductor of the circuit and that is connected to the neutral or grounded conductor of the system

Background
In the US, residential dwellings are often supplied using a 120/240V single split-phase system.  This system consists of two ungrounded (hot) conductors, and a grounded (neutral) conductor (which is a center tap on the transformer).
Multi-wire branch circuits
A Multi-wire branch circuit takes advantage of this, by extending both ungrounded conductors and sharing a single grounded (neutral) conductor. So instead of having two circuits requiring four conductors, you can have two circuits with only three conductors.
How to recognize them
Multi-wire branch circuits will typically be supplied by two adjacent breakers in a panel, and will often be fed using 3-wire plus ground cable.
If the circuit was installed properly, the breakers handles should be tied together or a double pole breaker will be used.
What to worry about

If you're working on the circuit, you'll want to make sure both breakers are off.
You should never disconnect the grounded (neutral) conductor, while the breakers are on.
You'll have to take care installing GFCI devices, as it can be a bit tricky.
You'll have to make sure the ungrounded conductors come from separate legs of the service.