A volume is simply the cubic area of a container multiplied out. An open, empty bottle of beer will have 1 volume of gas in it. At three volumes, it will show pressure in relationship to the temperature and molarity of the gas in use.
For brewers, the gas in question is CO2. Since CO2 is created in fermentation, a resting beer will have about 1 volume of CO2 dissolved in it, until it's degassed by heat or agitation.
The recipe has given you a precise target, leaving the method up to you. Nearly all carbonation calculations resolve to volumes, because this is reasonably universal and independent of batch size. There are carbonation charts for force carbonation that will show you the pressure required for the temperature of the beer. If priming sugar is used, the amount needed for volumes in a particular batch size is also easily calculated. BeerSmith's carbonation tool will calculate both.
