The Engineer:
The Engineer is a more tactically-oriented class for those who favor defensive postures and entrenchment.
His primary weapon is a pump action shotgun which does low medium-range damage and good short-range damage.
His secondary weapon is a grenade launcher which does moderate damage to an area.
His traversal tool is a "platform gun" which fires foam balls that explode on contact into a flat platform. The platform can later be upgraded to reduce fall damage if you land on it, and that can be important.
His special ability is to build a turret. The engineer only has one turret to start, though he can purchase an upgrade later after levelling up to get two. If the engineer does not remember to pack up his turret when he moves, he will have to wait for it to "hover" to him to catch up to the new placement when he tries to built. The turret does moderate damage at all ranges and serves as an excellent early warning/overwatch system for incoming enemies, especially grabbers and other sneaky types.
The engineer excels at guarding chokepoints and providing security in areas where "grabber" type enemies are a threat. He struggles with enemies at long range, however, and his turret is not smart enough to pick the best targets at all times, nor does it particularly aim for vulnerable spots. The shotgun is an excellent weapon at close range, but for best effect, headshots are a must - the shotgun can 1-shot most common enemies at point blank range if you aim for the mouth.
The Driller:
The Driller is a short range powerhouse which makes any given mission much easier. Most groups will find their lives much easier with a good driller among them.
His primary weapon is a flamethrower that does good damage at short range, and can be upgraded to do moderate damage at medium range. It can also be used to set the ground on fire, which causes damage to enemies that run through the fire. This weapon can also be upgraded to a "Freeze-thrower" after leveling up, which does light damage but is an immense help - it can freeze enemies solid, which makes them take extra damage from all physical attacks, negates armor, and is an instant-kill against flying enemies as they shatter when their frozen body hits the ground from any height. The flamethrower, however, can be used to ignite poison gas, burning it into harmlessness (abarring the explosive combustion effect).
His secondary weapon is a semiautomatic pistol that does low damage at all ranges. While not a particularly good weapon, it at least allows him SOME manner of long range attack, and is a fallback weapon for when the flamethrower runs out of ammo. It can later be upgraded to a charged plasma pistol that fires somewhat slower-moving projectiles that do AOE damage on impact, but I've found it to only be marginally more useful because it is extremely difficult to hit a moving target at range.
His traversal tool is his drill arms. The environment of Hoxxes is 100% destructable. Any class can use their pickaxe, but the drill arms are IMMENSELY faster. Simply by walking with his drills on, the driller can create a large tunnel through terrain, though he must be careful not to let his drills overheat (takes about 10 seconds) or they will be rendered unusable for a little while.
His special ability is remote-detonated C4 satchel charges. These bombs do immense damage at short to medium range (and range and damage can be upgraded), and also destroy terrain, but must be used carefully - the charges can easily kill the driller or his friends.
One of the nicest things about the driller is that he makes extraction to the escape pod simple - no longer do you have to run panicked through mazelike corridors hoping to make it to the pod before time runs out - the driller can just point in the pod's direction and start drilling.
The Scout:
The Scout
The Scout is a high mobility class that can really make life easier for his teammates, but is also very dependent upon them for survival.
His primary weapon is a fully automatic assault rifle that does medium damage at all ranges. It's an excellent workhorse weapon for all situations.
His secondary weapon is a double barreled shotgun. Similar to the Engineer's primary weapon, this does good damage at close range but quickly becomes useless the farther away the target is. Since it is double barrel, he can fire twice in quick succession (or with a later upgrade, fire both barrels at once), but the reload time is longer than the Engineer's pump action per-shot.
His traversal tool is a grappling gun. Using the grapple, the scout can move himself almost anywhere, at rapid speed. Paired up with an engineer, this makes grabbing minerals or other items off the ceiling of a cavern much easier and faster. However, the scout cannot use his other weapons (not even his pickaxe) while the grappling gun is deployed, so many novice scouts frequently fall to their death from high ceilings since they didn't realize they couldn't dig themselves a handhold once they got there.
His special ability is a flaregun. While everyone gets throwable chemical flares, the light from the Scout's flare gun is MUCH brighter, lasts MUCH longer, and has the advantage of being stuck into whatever terrain the scout shoots it at. The underground tunnels of Hoxxes are a dark and there are enemies everywhere. It makes everything much easier if you can see them coming. For fighting a swarm in a large cavern, a couple shots from the flare gun makes everything MUCH less dangerous.
The Gunner
The Gunner is a high-damage "tank" class whose specialty is dealing out punishment and protecting the others in the group. He's much more limited utility in situations that don't call for shooting things, but when a skittering horde of alien bugs comes swarming down a corridor, you'll be glad for his minigun.
His primary weapon is a minigun that does low damage per shot but fires VERY quickly. Fortunately he also carries lots of ammo for it. It does good damage at all ranges, though accuracy at long range can be a problem. This weapon can later be upgraded to an Autocannon which fires slower but does more damage per shot and also destroys a small amount of terrain.
His secondary weapon is a revolver which does good damage at all ranges, but fires very slowly and must be reloaded often. Many gunners like to use the revolver to snipe at long range, as headshots or other weapoint shots with the revolver deals excellent damage.
His traversal tool is a zipline launcher. Sort of a poor-man's grappling gun, but it becomes a permanent fixture - it creates a zipline from where the Gunner is standing to where he is aiming, and any member of the team can then use the zipline. Good for reaching distant ledges or crossing gaps, or keeping off the floor when it's covered in alien teeth and claws. However, ammunition is fairly limited and the angle of inclination at which it can be fired is very limited.
His special ability is a throwable shield generator. This generator creates a temporary "bubble" of shield that dwarves can enter but enemies cannot. This can often be a lifesaver if a swarm is overrunning your team, because it will push them back and give you room to breathe. The duration per use starts out pretty short, but can be upgraded as you level up.
By far the most straightforward class of the four, but many argue also the least essential. If having a gunner means you miss out on having an engineer or driller, than it might not be a good deal - and the scout can put out almost as much damage with much more mobility and utility. But don't discount that shield generator... it can really save your bacon when things are going sideways.