Broken Crank Arm - Chainring Connection
What you have is a 'one-piece' or 'Ashtabula" crank. Info can be found on this Sheldon Brown Page. They are called one-piece because the crank arms and axle are made from a single bar of steel. 

The chainrings are not fixed to the crank axle and can rotate on it, instead there is a 'drive pin' that sticks out of the right hand crank into the chainring spider. That's what has broken off your crank. From the picture, I think it was press fitted into the crank and worked loose.

I did some quick googling and found that replacement arms are available and inexpensive. Your chainrings do not look worn so you will be able to reuse them (although they could use a good cleaning &#x1f609;). 

If you are not experienced with bike repairs a local bike repair shop will be able to handle replacing the crank easily.

Update:

Park Tool has a good video on disassembly and servicing the bearings and re-assembly.