Damage depended on a variety of factors. Specifically what type of bullet/shell played a big part, as did the construction of the target aircraft and where specifically the aircraft was hit.
First, let's look at ammunition. Aircraft weapons during WW2 generally fell into one of three categories - rifle caliber machineguns, heavy machineguns, and cannons.
Rifle caliber machineguns were generally chambered in the same caliber as the country's service rifle (30-06 for America and 8mm Mauser for Germany, for example), and due to the small size of the bullets fired, generally were limited in the damage they could do. Tracers and incendiary ammunition could allow for more damage to be done if it hit the right places, but in general rifle-caliber machineguns were going to be most easily stopped by any armor present and limited to damaging whatever they've directly hit. Heavy machineguns were most common on American fighters (in the form of the .50 caliber AN/M2), although we'd see them show up elsewhere on occasion with designs like Germany's 13mm MG 131 and Italy's 12.7mm Breda-Safat.
Heavy machineguns were a middle ground between cannons and rifle-caliber machineguns, generally offering a greater potential to cause damage than a rifle-caliber gun while usually being faster-firing than a cannon and allowing the aircraft to carry more ammunition than a cannon. The Breda-Safat is an outlier for those last two points, though - it was notorious for having a very low rate of fire and Italian fighters carrying them often had very low amounts of ammunition. Ammunition for heavy machineguns still was generally too small to fit any useful amount of explosives, however, so again we see ammunition being limited to tracers and incendiary rounds at best. But, with more energy behind the projectile, they were less susceptible to being stopped by armor.
Cannons would come in two general classes - light (~20-23mm) and heavy (~30mm and up). The big selling point for aircraft cannons was the large size of their ammunition allowing for the use of explosive ammunition, with larger calibers being exceptionally devastating in the event of a direct hit. Though cannons in general tended to have lower rates of fire than machineguns, light cannon could still manage respectable rates of fire that often was equivalent to that of ubiquitous heavy machineguns like the AN/M2. Heavier cannon tended to have lower fire rates and muzzle velocities, significantly impacting accuracy of aerial gunnery, but had the advantage of being able to destroy even the sturdiest of aircraft with very few hits. For both classes of cannon, they've got the tradeoff of lower ammunition and larger size. Nevertheless, all countries by the end of WW2 were trending toward reliance on cannons for aircraft armament.
On the receiving end, damage depended on where the aircraft was hit. In a typical WW2 aircraft, the most immediately important parts are going to be the pilot, fuel tanks, engine (and any cooling system), and controls. Most combat aircraft had at least a small armor plate protecting the pilot's back and armored glass in the windscreen, though this wasn't always enough to protect the pilot, especially if they were up against heavier caliber aircraft guns.
Fuel tanks had the potential to be very vulnerable, especially to tracer and incendiary ammunition, but the introduction of self-sealing fuel tanks by most powers would greatly reduce the immediate danger of a hit to the fuel tanks. Japanese aircraft in particular were notorious for being vulnerable here, and the use of self-sealing tanks is usually cited as a major advantage US aircraft had over the Japanese early in the war.
Engine vulnerability depended very much on the type and layout of the cooling system - inline engines with liquid-cooling systems had the potential to be very vulnerable, while air-cooled engines could be incredibly durable. The P-51, for example, had its radiator relatively exposed on the underside of the aircraft, so a well-placed shot from the underside could potentially cripple its cooling system and kill the engine. The Fw 190, on the other hand, was powered by a radial engine that on at least one occasion could keep running despite an entire cylinder head being shot off.
Controls had the potential to be very vulnerable, but they could be difficult to intentionally hit. In WW2, control was provided by a series of wires extending to each control surface, and thus severing these wires could prevent the pilot from actuating the controls. Because they extended through the tail and wings, they weren't feasible to armor adequately, so in theory a well-placed shot from anything could sever control to an aileron or elevator. However, small wires in the aircraft are impractically small for an enemy pilot to aim for, so they were more resilient than one might imagine.
The construction of an aircraft could also make a huge difference in the damage done. Most aircraft in WW2 were of semi-monocoque construction (rigid shell strengthened with stringers), but there were notable examples of fabric-skinned frames (Hawker Hurricane being perhaps the most famous). Fabric skin had the unusual benefit of providing next to no resistance to enemy fire unless something inside the aircraft was hit, which, according to some anecdotes from the Battle of Britain, could be life-saving when the aircraft was the subject of cannon fire. If the shell never hit any of the structure underneath, the fuse might not be set off, and the shell would punch a nice 20mm hole through the aircraft and pass through relatively harmlessly. The stiffer skin of a semi-monocoque design wouldn't have this niche benefit, but aerodynamic and structural advantages outweighed this. Although aluminum was the most prevalent construction material, wood was also common, and in some cases (LaGG-3 stands out), wood structures were susceptible to shattering under impact.
On the extremes, you could have a situation where an aircraft comes home with thousands of holes in it and large chunks of seemingly critical pieces missing, or you could have a single unlucky bullet kill the aircraft.