Sprite

In the context of sprite animation, a sprite can be thought of as any 2D image that either comes from a video game, or is modeled after those (sometimes called a "custom sprite"). "Image" is the most general concept that fits here, but typically, this refers to pixel art. Throughout this wiki, you may see "asset" instead of "image", though that is a more generic term that can apply to audio, text, etc.

The definition above is loose enough so as to cover graphics in games like Angry Birds, for example (The Spriters Resource seems to agree with the definition laid out here). In this particular example, the fact that they're stored as bitmaps is misleading—they would still count as sprites even if stored in a vector graphics format, at least in this context.

Bits
Also used in confusing ways when it comes to sprite animation.

For our purposes, a bit can be thought of as a device that can store (or some other representation of) one of two possible values, "0", and "1". Within a computer, bits are commonly put together in power-of-two-sized groups (8, 16, 32, 64, ...), though arbitrary group sizes are perfectly reasonable, and are accommodated for in some way on real computer systems. Bit groups are normally meant to represent one entire value, with all the possible combinations of "0" and "1" interpreted in a way that makes sense for some given application and value range (for example, representing a whole number or "integer").

When it comes to CPUs, a denomination such as "8-bit", "16-bit", "32-bit", or "64-bit" typically refers to the size of a "word". The concept of a 32-bit (or 64-bit) program is closely related to this.

For images, bit depth (or color depth) is either the number of bits taken up by a single pixel, or the number of bits taken up by each color component (red, green, blue, or alpha, for example). When editing images, creating pixel art, etc., this is what you're dealing with.

Most of the time, though, when talking about sprites, "8-bit" (and so on) refers to a video game console generation, which is largely a market segmentation concept (one example of a mismatch between marketing and the technical specifications of a console would be the TurboGrafx-16, which, despite being released in the "16-bit era", makes use of a 8-bit CPU).

Spriter
A spriter is most commonly used as a synonym for pixel artist.

Other uses of the term
Similar to the term "sprite animation", "sprite" may be used to refer to ideas quite different from the one detailed above. In game development, for example, it would typically refer to some hardware-accelerated image display component, usually meant for dynamic purposes—like the player or enemy characters, bullets, special effects—in contrast with a more static component, meant for background imagery. These days, it's more likely to be used in a way closer to the definition above (that is, any image).