A spillway is an alternate path water can take around a dam when the water level gets too high (after heavy rains or flooding, for example). This prevents uncontrolled overflow over the top of the dam and potentially dangerous pressures from building up behind the dam that may be in excess of the dam's structural design capacity.
A typical spillway takes the shape of a waterslide with a curb in front of it. The height of the curb determines the height the water needs to reach before it overflows into the spillway. Sometimes they'll have features built in to help direct and control the flow of water.
The most dramatic type of spillway is called the morning glory spillway. Rather than being a waterslide type of design, a morning glory spillway looks like an enormous funnel. The top of the funnel flares out, normally above the waterline, on the reservoir side of the dam and empties out at the bottom of the other side of the dam. When the water level is normal, the top of the funnel is above the waterline and nothing falls in.
When the water level gets high enough to cover the top of the funnel, however, water surges into the funnel at an enormous rate. It looks for all the world like someone is draining an enormous bathtub, and the flow of water into the funnel resembles the morning glory flower it's named after.
The largest morning glory spillway in the world is in the Monticello hydroelectric dam in California. Pictures are available by searching the internet for Monticello spillway, just look for the enormous hole in the water.
I wouldn't recommend swimming near it.