Okay, having been a semi-professional graphic designer for over a decade now, I feel I have some authority to speak here.
JPEG - I use this format for photographs. I generally use a compression rate of somewhere (using Photoshop) of 8 to 10, depending on the size of the image. The higher quality I can squeeze out, the better. I don’t really like JPEG, but I know it’s the standard if I want people to be able to view my photos, so I use it.
PNG - I use this for images that have a lot of flat color. I do a lot of line-art, since I’m red-green colorblind (Deuteranomaly), so it works out perfectly. The transparency on this is ideal because it supports Alpha transparency, which means you have have blended transparency as opposed to sharp-edged transparency like GIF gives you. Plus it’s a more open format, so I don’t feel bad using it.
GIF - A dead format I detest, have always detested, will always detest. I used it only out of ignorance in the earliest days of my career.
MNG - A sadly unadopted standard. The images tend to be ridiculously large, thus making Flash a more viable alternative.
SWF - Shockwave File // Flash. It’s got it’s good points, but for simple animation I’m positive the open-source community can come up with a better standard. Vector-based imagery is already very possible (SVG being the standard), and not everyone that wants to display animation wants to have to shell out many hundreds for Adobe’s Flash suite. I’d love to see an open-source FireFox supported vector animation format.
Anyhow. What’s been said before me is pretty much in line with what is known about graphics in the industry. High-quality for-print images tend to be TIFF, low-quality for-web images tend to be JPEG. BMP has no business being used, and PICT, the Macintosh format, is already basically obsolete since Apple mostly uses JPEG/PNG when using their own internal imaging.
In all, graphics formatting needs to grow, again. It has not kept apace with technology. The mitigating factor has been web browsers, pure and simple. With FireFox basically leading the way, if they adopt something that is put into wide-spread use, it should be adopted by the other browser formats in short order. In essence, the onus is on the FireFox team whether they like it or not. Such is the burden of taking on the Big Blue E.