PNG, like GIF, supports lossless compression. This means that unlike JPEG, bits of image are not lost or thrown away when image is decompressed. This compression scheme is public domain and improved over algorithm used by GIF.

Images in PNG format may be made partially transparent. This is necessary for displaying good looking graphics on a web page.

Finally, PNG supports color depth of 24 bits or greater. This makes it a much better format than GIF for images which must have lots of colors.

An important thing to remember about PNG format is that color pallet is contained within image. Thus, it's important that pallet be made as small as possible, containing only colors that are necessary to display image. Most graphics programs should handle this detail for you when file is saved.

Note that JPG is still superior for large images because it's compression scheme is better suited for that purpose.

In addition, PNG design does not include support for animation. Thus, GIFs must still be used for that purpose.

Most of modern graphics editors support output in PNG format. These editors include Paint Shop Pro, which is my personal favorite.

Personally, I think it's a little early to go and replace all of your graphics with PNG graphics. Support within different browsers is still very new, and you would be depriving many of your visitors of your graphics by including PNG images. However, in a few years I expect to see this format more and more.