Gut is a slang word for your stomach and intestines or just that part of your body - the man in the picture has a large gut.
When you are preparing a whole animal to be cooked, the process of gutting it is the process of removing its internal organs, including its stomach and intestines.
Gutters are channels that are used to carry water away to somewhere else. Gutters can be found around the edges of a roof, to catch the water and take it to storage or into a drain. Gutters are also found at the edges of roads, and sometimes also in the center of a road, to carry water away from the surface of the road.
