The majority of monster attacks deal only "normal"-type damage. However, some attacks deal piercing, elemental, or shadow damage. Piercing is the least common; I've only really encountered it against the larger lichens - the ones big enough to fire spikes. Also, floor spikes deal piercing (and normal at the same time). Elemental is probably the most common; many/most projectiles seem to be elemental, which includes gunpuppies, for example. Shadow seems to be limited to undead and fiends.

You can get a clue as to what sort of damage a monster's attack will deal by looking at the telltale glow that appears in front of the monster right before it attacks. Red means "normal", yellow means piercing, purple means shadow, and green means energy. Some attacks deal two elements - typically normal, plus a second element. You can also tell what sort of damage you've taken after you've been hit, by looking at the colour of the health bars that fall off of you.

I recommend energy defense for sure. In particular, I always use a shield that has energy defense, as most of my blocking is done against gunpuppy spam and other projectiles. Shadow is relatively optional, but if you don't have any, avoid levels with undead and ghosts. Piercing isn't terribly important, but it doesn't hurt to have it (especially if you're careless around spikes).