A sprite can be a single image, or an animated series of images, that represents a game object such as a character, boss, or stage enemy. They are often laid out in grids on a Sprite Sheet so the game engine can easily identify and load the animated frames.
The size of a sprite used to be dictated by the hardware required to draw it, but modern games don't have that restriction. Still, most retro games still adhere to a grid system when designing them. For example, on a grid system of 16 pixels, a character could use 16 by 16 pixel dimensions, or may use 16 wide and 32 pixels tall. Some animations, like attacks, may use 4 grid units in a square of 32 by 32 pixels.
Some games use 16 pixel units, some use 32, and some games deviate from that format and use 20 pixel units. The main thing to keep in mind is that whatever standard *your* game chooses, be sure to stick to it.
Tiles are similar to sprites, but are meant to serve as the building blocks (sometimes literally blocks) of the game world you're creating. Tiles are laid out in a Tile Set for efficiency, and the game engine can assign properties to them (such as whether a character can walk over them or collide with them.)
All tiles on a Tile Set should be the same dimensions, and are usually all square. Larger objects can be made by combining several smaller tiles to make a larger object. (Like the cloud below.)
Tile Sets can be laid out with no spaces between them, or some game engines may recommend a "gutter" or "bleed" area between the tiles to prevent tile tearing in graphics intense situations. See illustration for examples of how tiles can be used to make game levels.
Splash art is often a full-screen graphic, used at a title screen, loading screen, or cut scene. Splash art usually does not need to conform to any pixel dimensions, and the art style of these can be wildly different than in-game graphics due to more color or resolution available. Your splash art should conform to the style and color scheme of your game graphics as much as possible to maintain a common look and feel.