In the context of firearms the word caliber (also "calibre") indicates the approximate diameter of the barrel at the muzzle - and by extension the diameter of the round fired.
Calibre is usually expressed in inches (or fractions thereof) or milllimeters, depending on jurisdiction of origin. In shotguns the calibre may also be expressed as "bore" or "gauge" which is an archaic measurement of the number of spherical balls of that diameter that could be cast from a pound of lead1. Calibre may be nominal (that is, the stated calibre of a weapon may be rounded up or down a little from what appears on the machining diagrams), but someone should have already taken account of this when assigning the ammunition.
As a rule of thumb, the larger the calibre, the more powerful the weapon (for equivalent classes of weapon), although your mileage may vary considerably.
Early ordnance grade weapons may also have their calibre expressed in 'pounds' (e.g. the 12lb Napoleon cannon), much as for a shotgun, but in this case the shot in question will be cast iron and the 'pound' used to cast the shot inquired into very closely (as practically every nation had at least one 'pound' or equivalent measure that varied from everyone else's). This terminology continued into the mid 20th century (especially in the British Forces) but had less and less to do with calibre as ammunition became more complicated over the years.