A sprite is a 2D image in a video game made up of pixels.
What is a sprite?
Although the term is most commonly used to refer to game characters or moving objects, the technical definition of a sprite generally means any 2-dimensional image in a game. This includes static objects, backgrounds, terrain, even UI elements such as buttons and textboxes. In order to qualify as a sprite, the image must be stored as pixel data within the game. For example, programmatically generated effects ([need an example]) are not sprites; however, effects that were stored in the game's data as images, like an explosion animation, are sprites. Sometimes there are mixtures, such as a particle emitter, in which case the individual particles it emits are sprites (but the emitting itself isn't).
Animations and Frames
Most sprites, such as characters, enemies, and other moving objects, have animations. Animations are made up of images being displayed in sequence in such a way to show movement (or other non-static features such as sparkling). These individual images are called frames.
So for instance, one would say that Mario is a sprite, and one if his animations is him walking. This animation is made up of multiple frames of him, each being displayed after the other to depict him walking.
As an example, here is Mario from Bowser's Inside Story.