Some may be saved with clipping paths. I'm not sure how that works, but you may want to look into it. I think artists/photographers usually indicate that in the description, so try including the phrase "clipping path" in your searches for icons.
Maybe I'm stating the obvious, but in Photoshop use the "crop tool" (key shortcut "C")and save each individual icon in a new separate file, or if you are placing the icon images in either Illustrator or Indesign, just marquee select individual icons and copy (control+C) and paste (control+V) into those programs. There really is no "automated" way, since you need to manually select the area you want to use...sorry if I'm being simplistic, but I'm assuming you are not familiar with the programs at all, and I know they can be overwhelming to a non-user.
as long as the background is the requested background and you don't need to cut around edges .. I woud use Adobe fireworks - its great to crop icons to use for web as you can export them optimised. Newer versions of photoshop is very good with this too.
If the image is an illustration, you can ask and download the vector file in 'additional format' area. The example is illustration, and a vector version is available, where in the user can use each icon separately.