The drivetrain is everything from the pedals to the rear hub, including crank arms, chain rings, chain, front and rear derailleurs, the rear cogs (cassette/cluster) and the freewheel or freehub.

Shifting issues

Derailleur hanger bends
Symptoms include rear shifting “skipping” or refusing to go into or stay in gear. Most modern hangers are designed to do this in order to protect the derailleur.
Front derailleur breaks
Hard to do, but if you bend the guide plates your chain won’t stay on the chain ring. Tips for reparing or removing the derailleur.
Rear derailleur breaks
Normally caused by a stick in the spokes or a trailside rock. It’s unlikely you’ll be able to fix this on the trail, so it’s time to go single speed!
Gear cable snaps
The shifter feels wobbly, the chain is in the biggest cog and you have three more hills to climb before getting home.

Chain issues

Chain skips
If the chain is sliding over the gears rather than meshing with them, either the chain or the gears are probably too worn.
Chain breaks
Trail pixies must be the biggest cause of broken chains, because everyone I know swears they take good care of theirs.
Chain “sucks” around chainring
Chain suck is the term for the situation where a chain ring doesn’t let go of the chain in time, leading at its worst to the chain getting wrapped around the bottom bracket area.
Chain shifts into rear spokes
This invariably happens when you are standing up to crank up a short incline. You shift down, and the next thing you hear is a “clunk” as the chain over-shifts

Broken drivetrain parts

Chain ring loose
Shifting issues, bad noises, long-term bike damage.
Chain ring tooth broken
A broken or bent chain ring tooth will cause havoc with pedaling and shifting.
Chain ring bent
This typically happens to the outer chain ring when you get overly ambitious with logs and rocks.
Crank arm loose/falls off
If your crank arms “flop” with every pedal stroke, or if you can wobble them from side to side, fix them now before they become irreparable.
Freehub freewheels in both directions
It’s the strangest feeling to pedal forwards and not get anywhere. If it’s just started happening, you may be able to temporarily fix it.
Cassette (rear cogs) loose
A loose cassette causes several issues. It tends to rattle around, and makes shifting difficult in both directions.
Pedal breaks
Normally pedals don’t break so much as fall apart. So long as you have the bits, you can fix it. If you’ve lost the parts, you can still ride out.
Drivetrain gummed up
Mud, ice and snow gum up the chain rings and cassette and make shifting difficult or impossible.