The different conventional damage types refer to different classes of weapons; the weapon description will say whether it does slashing (sword, knife), crushing (club, walking stick) or piercing (spear). Shadow and bone do non-conventional damage (sort of magical / spiritually based), though the weapons themselves are just derived from conventional weapons (bone sword, shadow bow, etc).
Some creatures are stronger or weaker against different damage types. Conventional damage is a good all-round damage type that works pretty much the same on everything (there are of course variations, but resistances to slashing, piercing and crushing tend to be fairly neutral).
For the majority of the first act there are lots of skeletons around, which are a bit more vulnerable to crushing damage. Until I reached the citadel I had my hero use crushing weapons (a club or 2 handed walking stick) most of the time and a bow for tougher fights, then I switched to archery full time. My DK continued to use slashing damage (1 or 2 handed swords). Playing again I'd have the DK be a bow archer to concentrate on agility, as he doesn't need a lot of strength for armour (his armour class is based on character level).
Particular creatures can have a much wider range of resistances with other damage types; for example, spiders have a 100 resistance to poison, but -100 to ethereal damage.
Specifically for shadow weapons (which are covered by a separate skill from the slashing expertise), most creatures in act 2 would not be too different. Ghosts (for one) would be easier to kill, since they have a shadow resistance of -80, while fire demons would be harder to handle, with a resistance of +100.
Playing a warrior / archer combo, I collected all the different arrow types thinking if I had a problem with a particular creature I could switch to something they were vulnerable to, but I ended up just sticking to conventional damage (normal arrows, then power arrows, and splitting arrows came in handy once). If you don't mind switching weapon types to match target vulnerabilities for various areas (or specific bosses), it can make things easier.
