You die when your HP drops below 0. When you die, some of your belongings will drop to the ground and you will lose about 1/3 of your money. For characters level 5 and above, there is also a 25% chance that your skills will decrease by 0.5 points.
Additionally, the majority of the items which you lose upon death will be left at the exact spot of your death. The probability of dropping your equipment dying on the local map is lower. Dropping equipment on maps that respawn (like towns) means that your equipment will be eliminated within a few days of your last visit. Dropping equipment on maps that are not saved at all, and are re-randomized with every visit, like the Puppy Cave, means they are lost forever.
Your score is dropped to 1/10th of what it was every time you die; unless you're going for a high score, this isn't a big deal.
None of these penalties apply if you die during the Hunting, The Harvest Time or Party Time! job Quests, or while in the Arena. Presumably, this is to prevent the problem of losing equipment forever in these always-re-initialized maps, and it was simply never applied to places like the Puppy Cave. Instead, you will lose points of Will and Charisma as a punishment for dying (This was added in version 1.22 to prevent players from abusing death as a means of evading certain negative Status Effects, such as anorexia). There is no such penalty in the Arena, however, where you merely lose Arena Rank and Fame.