Backend (also spelled Back End, Back-End) refers to Backend Development. The backend's responsibilities include storing data with the database, serving data to the frontend, user authentication. Due to the separation of the frontend, the backend is not limited to any particular language or technology stack. Examples: NodeJS, Rails, Scala, PHP, and more.
Full Stack refers to Full Stack Development. A full stack developer is one who has expertise in both frontend and backend development, and can adeptly move "up and down the [technology] stack".
Imagine a restaurant. The waiter (frontend) takes and sends your order to the kitchen. The kitchen (backend) prepares your order. When the kitchen receives your order, it prepares it, and the waiter deliver the order to you.
Fun fact: A restaurant kitchen is called "back-of-the-house" and the server crew is called the "front-of-the-house".