Under-inflation puts too much pressure on the tire’s sidewalls. You can tell when a tire is underinflated, because it looks like it’s bulging outward.
This is a big problem, because under-inflation TRIPLES the chances of a blowout - but it can also cause longer term tire damage, creating weak spots that could fail later even after the tires are back at the right pressure. Tire wear will increase, along with fuel consumption, because it takes more energy for your car to roll an under-inflated tire down the road.
Over-inflated tires are trouble, too. The can cause blowouts on long drives - especially in extremely hot temperatures. They also impair handling because over-inflation means less of the tire’s surface is in contact with the road than the manufacturer intended.
You should never exceed the upper pressure limit specified on the tire.