Vector or raster?
Vector images are mathematically-encoded lines laid down as strokes and fills. They can be scaled up or down perfectly, and animations are typically accomplished through a skeletal-style animation system -- Many flash games are examples of this style. Adobe Illustrator, Inkscape, Flash, and AnimeStudio (I believe) are examples of vector art tools.
Raster images are 2D grids of pixels, typical of a bitmap or similar. Sometimes called pixel art or sprites. They can scale well taking into account the size of the source image and the range of sizes it will appear as on-screen, and can be combined with mip-maps for better scaling. Animations are typically accomplished on a frame-by-frame basis. Adobe Photoshop, GIMP, Paint.Net, GraphicsGale, and Cosmigo ProMotion are examples of raster art tools.
Some Raster tools are geared more towards high-resolution, high-color work (Photoshop, GIMP) such as photo editing -- but can also be used to create such images from whole cloth. Others are geared more towards low-resolution, low-color work (Cosmigo ProMotion, GraphicsGale) and often include built-in support for frame-based animations; these are great for the kinds of 2D sprite-based games typified by the SNES, GBA, or 2D arcade games.
I personally really like Cosmigo ProMotion for 2D pixel art in that retro style, which is what I like. I'm not a big fan of 2D vector art, so I've never looked at or considered those tools.