Gills are what creatures that live in water use to breathe. Fish gills are easy to see, between their head and their side fins. Other creatures have less obvious gills, that look like fine, tufty growths.
A goatee is a kind of beard that is only on the point of a man's chin rather than on the sides of his face too. A goatee has that name because it looks a bit like the kind of beard goats have.
Gums are the pink lining of your mouth around your teeth. When your gums get inflamed, it is called gingivitis. Keeping your teeth and gums clean and healthy helps to prevent gingivitis.
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.