Pop rivets is what you're looking for. If you live near a Harbor Freight, they have a cheap gun that's good enough for very occasional use. It comes with a small selection of rivets for about $10.
Rivets look like a mushroom with a rod sticking out the top. The stem has a bulb on the bottom end, and as the rivet gun pulls the rod up, the bulb wedges into the rivet head shaft and causes it to expand, trapping the rivet. The gun then breaks off the shaft at the rivet.