stiff gears and chain coming off while pedalling forward (hub gear)
There could be a few different issues:


It your Rear wheel is sliding around in the rear dropouts and you wind up loosing tension on the chain or screwing up the alignment of the rear wheel. It's really a common thing on bikes with horizontal dropouts, which I assume your bike has.  When you start to pedal, if the rear wheel bolts or quick releases are not tight, then the wheel can slide forward in the dropout, the chain comes loose and falls off. Try tightening bolts or quick release tighter or use a chain tensioner or tugnut to help prevent the axle from slipping.
Your chain is really worn and/or your cogs are very worn, this causes the chain to skip off the front chainring or rear cog. See Sheldon Brown's site about chain wear to diagnose this issue. 
You have a bad chainline.  Instead of going from the rear cog on the wheel to the front chainring in a straight line, it bends the chain diagonally, causing it to hop off the cog or chainring easily.  This may be due to spacers incorrectly placed on either side of the rear axle. Credit @DanielRHicks
The chain could just have too much slack, try pulling the wheel back and resetting the tension tighter.
Bent chainring causes chain to jump.  Spin the chainring without the chain and see if it 'wobbles', if so this could be causing your chain jumps and the chainring should be replaced.