Each element is tied to a certain enemy faction. and their shields is that element. using the element on them deals bonus damage and helps take down the shields much faster.
- Fallen - Arc damage, signified by blue shields
- Hive - Solar damage, signified by orange shields
- Vex - Void damage, signified by purple shields (exception: some Harpies have arc shields)
- Cabal - Solar damage, signified by orange shields
Most of the time you won't need the element to deal damage, but in a few rare cases I've noticed one area so far with Vex Harpies with Arc shields that take damage from Arc attacks, but resist all others (still do damage, just a lot less).
Your primary weapon will usually be Kinetic damage. Your special and heavy weapons can and should have an element. Preferably one of each element so you are prepared for any encounter.