Beam refers to the width of a boat. As with length, there are a few ways to define beam, depending on what you need the number for.
Units: Linear measure (usually metres/centimetres or feet/inches).
Beam overall (BOA)
The maximum width of the boat, excluding rub rails or lifeline stanchions that may protrude past the edge of the hull.
Beam waterline (BWL)
The width of the waterplane when the boat is loaded to the design waterline shown on the plans. On multihulls, BWL refers to the waterplane width of a single hull.
Beam centreline (BCB, BCL)
For multihulls only, the lateral spacing between hulls.
Which one to use?
For most purposes
Beam overall is the most commonly used description of a boat's width. It gives some indication of how spacious the interior will be, and how wide a slip needs to be to accommodate the boat. BOA, plus a comfortable margin, is the dimension used when determining whether the boat can pass through a canal or fit in a slip.
When comparing sailboat performance
For monohulls, BWL is the quantity of interest. Wider beam implies greater form stability, within a given family of hulls. All other things being equal, the hull with more form stability will be less tender and better able to stand up to a powerful sail plan. Making a hull wider, though, will generally increase wave-making resistance, and often increases wetted surface area- thus increasing frictional resistance. Finding the right trade-off is one of those "no right answer" optimization problems that make good design tricky.
Multihull performance depends on both BWL and BCL. The former contributes to the resistance; the latter contributes to the multihull's righting moment and therefore its sail-carrying power.
The effect of beam on performance is better understood by looking at the length/beam and beam/draught ratios.
When comparing powerboat performance
BWL is, once again, the appropriate quantity to compare. Since a powerboat doesn't need to stand up to a full press of sail, initial stability (due mainly to form stability, or hull shape) is more a matter of preference- although there are long-standing guidelines for what works well in a given situation.
If a hull is capable of planing, BWL becomes a very important factor. A wider running surface gives more planing lift, but also presents more area for the hull to pound on when bashing through waves.