Your gut is part of your digestive system — it's where food goes after you eat it, to get broken down into nutrients and waste. The expression to "go with your gut" means to follow your instincts, how you feel in your belly.
Your gut is quite literally your intestines, although the word is frequently used for any part of your digestive system, especially your stomach. Eating too much ice cream can give you an ache in your gut. When gut is a verb, it means to clean the internal organs out of a fish or other animal you intend to eat, or to completely empty a building. Colloquially, if you have guts, you're very brave.