a hand where you have 4 of the 5 cards needed to make a straight, but your 4 cards are not "connected" or in sequential order, so you need a single card in the middle of your straight to complete the straight. i.e. - you are holding 5 6, and the board shows 7, 9, 10. At that moment, you have a gut shot; only the 8 will make your straight. This type of hand is also known as an "inside straight draw" or a "belly-buster straight draw".
A Gut Shot is like filling an inside straight in poker. It's having the starter card fill in between the other cards in your hand and significantly increasing the point total of the hand. While there are hundreds of examples, probably the most famous Gut Shot is holding 4,4,6,6 in your hand and having the Starter Card be the 5 for a 24 hand.