GIFs make the smallest files for line art -- text, comics, and emblazons. PNGs vary depending on the settings. JPEGs make the smallest files for photographs and anything with gradations.
Here's an example of images of comparable quality in three different file formats. I used Photoshop to make these from an original Photoshop .psd file.
This is a GIF. It is set for 32 colors and is 8.5k.
This is a JPEG. It is set for high quality and is 35k. I had to make it high quality to get an equivalent in visual quality to the above GIF. Lower quality caused blotchiness around the lines and in the green color field.
This is a JPEG. It is set for low quality and is 5.5k. To get this to be smaller than the GIF above, I had to choose a low quality setting, which causes blotchiness. Look closely around the edges of the beacons. Compare it to the other images on this page to see the difference in quality.
This is a PNG-8 and is just over 10k. Note that Photoshop makes PNGs that are larger than they ought to be. In general, if I want to make PNGs I would probably not use Photoshop. In Photoshop, a PNG-24 was about as big as the first JPEG I made above, so I didn't include an example of PNG-24.
-- Wenyeva atte grene