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It all depends on what the image is!
If it's a few solid blocks of colour, such as a logo or a vector graphics, then a GIF is the most appropriate.
If the image is a photograph, then the best format you can use is a JPG because of the type of compression.
If you want to use an image with more colours than GIFs support and have transparency, then you will have to use a PNG. PNG is also for alpha-transparency, which means it allows more than one level of transparency. But beware, IE has problems with 24bit PNGs (alpha transparent images) and hue/color matching for normal PNGs. This can be fixed, but should be avoided.
You can read up on image formats anywhere you like, but here's a starter - Image formats.com
PNGs should also be fine if you're colour matching to background colours or GIFs, as IE (again) gets the colour slightly incorrect, but noticeable.
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