In the field of locomotives, adhesive weight is the amount of the locomotive's weight that's usable for traction; in other words, the weight that is carried on the driving wheels and not merely the carrying wheels, whether leading or trailing or other.
Counterintuitive it might seem at first, but the maximum possible usable hauling capabilities of a locomotive are entirely defined by this. The number of driving wheels is not relevant to the tractive abilities whatsoever; for a given adhesive weight, two driving wheels are as good as twelve. The number of driving wheels is the simple result obtained by dividing the adhesive weight by the maximum axle load permitted on the tracks the locomotive operates over as laid down by the railroad's civil engineer. If the adhesive weight is 100 tons and the civil engineer only permits an axle load of 20 tons, then the locomotive will have five pairs of driving wheels.
The one benefit of having more driving wheels, aside from reduced axle loading, is an improved resistance to wheelslip from slippery patches on the rail, since not all the wheels are likely to be on the slippery patch at once.
The relationship between adhesive weight and tractive effort (the pulling force available; the figures quoted are normally that available when stationary) is known by the term factor of adhesion. A locomotive with a high factor of adhesion will never slip, while a low figure indicates a locomotive that will require careful handling because its power possibly exceeds its grip on the track. Such a locomotive will use a lot of sand - the application of sand to the railhead being a well-tried method of temporarily increasing friction.
Obviously, the friction between wheel and rail varies depending on the materials used, their condition, moisture, contaminants on the track surface, etc. but it's generally considered a constant factor for the purposes of specification and calculation.
If maximising adhesive weight is always the goal, why do locomotives have unpowered load-carrying wheels? Stability is the general reason, because (especially in a steam locomotive) driving wheels carry a lot of unsprung mass, and are thus not really suited to leading the way at speed - they have a tendency to ride rough and even jump off the track.
Unpowered axles on a diesel locomotive generally exist because a locomotive needed more axles (to spread the load, or for a better ride) but there was no room to put in additional traction motors (which, especially early on, were quite bulky devices). This is the reason for the unpowered carrying axles in the middle of the trucks of many American express passenger locomotives, giving them a wheel arrangement of A1A-A1A, allied with the need, in that application, for power at speed, rather than maximal low-speed hauling ability.