There are a wide range of tasks that engineers can perform from repairing vehicles to explosives/EOD roles. It's important to note that ACE considers some units to be engineers and others to be normal. Some tasks can only be performed by engineers whilst others, like changing tyres, can be performed by anyone but engineers perform those tasks faster.
Vehicle Repair
Field Repairs
In the field an engineer can, by default, only repair 40% of the damage to a vehicle. If the engineer is carrying a toolkit this can be boosted to 60%. In order to fully repair a vehicle a repair vehicle/facility and repair specialist are required.
Repair Facilities and Vehicles
Vehicles and objects can be assigned to be repair vehicles and facilities, respectively, by the mission maker. A repair facility or vehicle will allow an engineer to fully repair a vehicle (as long as the engineer is also designated as a “repair specialist”) as opposed to the partial repairs that are completed without the vehicle/facility.
Changing a Tyre
Before long you'll find yourself in a situation where your vehicle has lost a tyre. Fortunately, all wheeled vehicles carry at least one spare tyre by default.
Unloading the Tyre
Find the vehicle's interaction icon, go to “cargo”. A new UI will appear. Choose one of the tyres and click “Unload”. The tyre will spawn as 3D object somewhere around the the vehicle.
Move the Tyre
Use the interaction key on the tyre to access the “Carry” option under the wheels interaction icon. Move the tyre close to the damaged wheel and drop it by left-clicking. Use care when moving the tyre as Arma Physics means that touching player with the tyre can injure or even kill them. Make sure to announce loudly that you're carrying what is, as far as Arma is concerned, a round, rubbery lightsaber.
Repair the Wheel
Look for the “Change Wheel” icon on the damaged wheel and select it. After a short time the wheel will be restored. Try to not drive into a rock this time, you clutz.
Repairing a Track
Fixing a track works almost identical to replacing a wheel. You unload the track like you would a wheel, pick up the track segment and move it to the broken track. You then repair it by using the interaction key just as you would with a wheel.
Explosives and EOD
Placing Explosives
To place and detonate explosives you need certain pieces of equipment. At a minimum you need an explosive but you will likely want a detonator to allow you to remotely detonate the explosive instead of using a detonator.
Disarming Explosives
To defuse an explosive you'll need the Defusal Kit. Interact with the explosive and select the “Disarm” option. Certain explosives will detonate on defusal.