That entirely depends on what you are using the graphics for. There is very little use for GIF files these days as PNG has replaced it for most tasks.
JPG files are only suitable for large images with lots of textures and detail. The images in JPG files are compressed using a lossy format, so some loss in quality is observed on converting to this format. However the loss of quality is compensated for by a huge drop in file size for images containg lots of detail. Photographs are almost exclusively published online as JPGs for this reason.
PNG files are very effective for displaying text or very small photos. The file format is much newer than both JPG and GIF and as such features a much better compression method. PNGs are saved in a lossless format so there is no loss in quality on converting to this format. Although there are options to save in PNG8 which features a colour pallete like GIF, if you convert an image with 20,000 colours to a PNG8 file with only a 256 colour palette you will notice a considerable drop in quality, however if you keep the existing colour palette then there will be no loss in quality at all.
GIF files are useful for creating small animations which repeat over and over. They also allow for transparencies by making one of the colours in their (maximum) 256 colour palette transparent.
The PNG format can also create transparencies and supports an alpha layer which allows for semi-transparent portions. However support for this format is not available in <IE6 so it isn't much use except for in particular circumstances. Once the need to support IE6 is gone, alpha level transparencies in PNG format will be very useful.
Another format you can (in theory) use is MNG, which is a video format similar to the PNG format for images. However there is minimal/zero support for this format in web browsers so it is virtually useless. Adobe Flash is the best option for videos on the web these days.
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