Dampers, also known as shock absorbers (or if you're replacing a set, you may need to put the word strut in your search because in some cars they are integral with the strut) work in conjunction with the springs (and roll bars where they are included) to help manage and control a vehicle's behaviour.
When the damper is compressed it's called either bump or bound, and when it extends again it's called rebound. In many cases dampers will have clever internal valving to separate this behaviour over sudden road irregularities (called fast bump and fast rebound) from that of the overall and much slower up and down movement of the chassis in relation to the undulations of the road or a smooth change in direction (controlled by the regular bump and rebound).
The fast settings are mainly needed for things like potholes on the street, or kerbs in a race car. Therefore, if you're judging a vehicle's comfort levels over bad bumps, you are, in part, judging its fast bump and fast rebound settings.
The fast bump needs to be fairly compliant (or soft) so as to quickly react and let the wheel bound up without upsetting (transferring that bump to) the chassis too much.
Now, remember that point about the damper working in conjunction with the spring, which (as a result of that pothole or kerb) has now been compressed too and is trying to extend back out again. On the far side of the kerb or pothole the suspension then needs to let that wheel down again, but not so forcefully that it upsets the chassis anyway (or wants to keep bouncing or oscillating), so the fast rebound rate will be a bit firmer (stiffer) to control that spring extension.
However, if the fast rebound (or rebound) is too stiff (and the bumps just keep coming) you get a condition called jacking down, which is where the wheel hasn't yet been allowed to extend to its normal position (relative to the chassis) before the next bump cycle and it now has less travel to work with. Keep this going and you'll run out of travel.
The slow (or normal) bump and rebound settings need to be a bit different to that. Not as soft for bump and not as stiff for rebound anyway, because there is far less energy being put into the spring the rest of the time you're driving around.
In conjunction with lots of other factors (suspension type and geometry, springs, anti-roll bars, total mass, the weight distribution, the diff settings and more) the dampers also influence how a vehicle seems to handle, and how much grip it has at each end when braking as well as when going into, through and out of a corner.
In competition vehicles, it's desirable for all four of these settings (bump, fast bump, rebound, fast rebound) to be adjustable, allowing engineers to fine-tune the vehicle's behaviour for a particular course or circuit.
Vehicle manufacturers do extensive testing to get the compromise acceptable and hopefully they get it right. Some countries will then put them through rigorous standardised assessments like the S-shaped double lane change Evasive Manoeuvre Test in Sweden to ensure they can dodge a sudden obstacle and return to their own lane without tipping over or crashing some other way.
One method to find a good baseline setting for bump in adjustable dampers (which can be fine-tuned later for different tracks) is to set them fully soft and send the driver out for a few laps, then firm them up a few clicks and send them out again. Keep adding a few clicks until the car seems to stop side-walking (understeering and/or oversteering caused by the tyres skipping across the road) and then back them off a click or two. If one end seems sorted and the other not, keep adjusting that end until it is. Adjusting rebound is similar; start soft and increase only until it rolls with a smooth controllable motion.
Think about what that means for a road car then if the dampers are old and worn. And you can also find that you've introduced a similar problem if you install lowered springs (which in most cases are also stiffer) because the old dampers will probably be a bit too soft to control them properly.
You therefore need to ensure your dampers are always in good condition if you want the vehicle to remain safe and controllable, which is vital in an emergency situation.