PNG - This is the lost brother in Internet file formats. Originally the Portable Network Graphics format was developed by Thomas Boutell, Tom Lane, et al as a replacement format for both GIF and JPEG - it natively supports all the characteristics of both formats, and then some, but it seems to have been forgotten in the push to make web browsers bigger, better, and more burly. Only recently have browsers begun supporting this format without the need for special proprietary plug-ins.
As mentioned above, the greatest advantage to using PNG is that it supports all the features of both GIF and JPEG. If you have text or line art, you can activate the GIF-like color reduction options and create a smaller file with no lossy compression. On the other hand, if you're saving photo-realistic images you can activate the JPEG-like compression schemes and save them that way, also with no loss during compression (unlike JPEG which varies from near-lossless to extremely lossy). In addition to these features, it also supports alpha channels for masking, gamma for color correction, and storing 48-bit color images (JPEG can only go as high as 24-bit).
However, there is a drawback to all this: PNG files tend to generally be larger than JPEG files as they almost always use a lossless compression scheme. (Though often times when used with the same color palettes as GIF files they tend to be smaller.) Additionally, while modern browsers do natively support this format, they don't necessarily support all of its functionality. Finally, PNG does not support animation, perhaps the only feature it does not share with GIF.
If you decide to go with PNG, you may find the results are not always what you expect.
