Table of Contents:
File Types
Background Colors
Color Limits
Frame Rate Conversion for Web Animations
'Reduce Colors to' settings (or 'How to avoid those weird dots')

File types
NEVER save a sprite as a .JPEG/.JPG file. They make your sprites blurry and completley ruin them. Here's an example of a Hiryu sprite from Marvel vs. Capcom saved as a .PNG file and a .JPG file. Each has been zoomed to 300% view and not resized by the art program. You can see how much damage is done even to the simple letters in .JPG.

.PNG : These are perfect for non-animated pictures that need transparent backgrounds and have less than 255 colors. They are also good for pictures with more than 255 colors that don't have need transparent backgrounds. They're even better than static .GIFs because of their small file size and easy use with earlier versions of PHP. They can also have extra effects, but only on certain web browsers.
.GIF : These are for animated pictures and can even have transparent backgrounds.
.PCX : This is a good format that doesn't blur the sprites. This is also the format that is used by M.U.G.E.N.
.PSP, .MNG and .PSD : These have the benefit of saving many frames of animation as layers that can be later used in some animation programs like Animation Shop.
.BMP : A good format for saving single frames. The problem is that each file takes up A LOT of space.
.JPG and .JPEG : NO! They ruin sprites.

Background color
Don't use black or white as a backgrond color of a transparent .GIF or .PNG file unless the color is crucial for an effect. Many times, people reduce the colors and accidentally merge the colors for the background and a similar color in the graphic. Instead of black or white, try using colors that very different from anything used in the graphic. Popular colors are Magenta (#FF00FF or 255/0/255 Green (#00FF00 or 0/255/0).

Sprites and graphic tiles on older systems and arcade games had limits in size and color that were more obvious than they are on more recent systems. You should try to learn and follow the color limits of a given style if you're trying to make sprites or sprite edits that seem to fit into another game. Sometimes they got around these limits by overlapping separate sprites that had different palettes. You'll see this in certain fighting game sprites like Wolverine's claws in X-Men: Children of the Atom, Sodom's swords or jutte in the Street Fighter Alpha series and Captain America's shield in Marvel Super Heroes. Thanks to Capcom doing that, you can 'cheat' by using extra colors on weapons or things that would be considered separate from the actual character when using those styles.

Each system also has it's own palette set or even multiple palette sets. Games colors will also differ depending on the style that the developers use too. If you want to imitate a game style, get a wide variety of sprites and copy their colors as much as you can. You won't want to make really bright colors for a game with moody, washed-out color tones because they won't blend in. You'll really want to pay attention to this when making fake screenshots.

It's not that you have to do this but working within the limits that the original game artists work in make things look more authentic. You may also develop a bit more respect for early sprite pioneers for realizing how much they were able to accomplish within the strict limits.

Sources and Reference: