Several have asked, so here goes:
Open the file. Use various selection tools as needed to select only the parts of the image you want to keep. Copy. Open a new doc, selecting the transparent
background option. Paste. Save as Compuserve gif format. Ta daaa!
Note that if you don’t use the gif format, and instead use, say, jpeg, the background will be filled in with a solid color again.
If you use the Photoshop format, it can’t open in other programs or show on the Web.
As for selecting areas, try the magic wand 
if you haven’t already. It is one of the best tools for this kind of selection.
Here’s a sample.
In this case I’d probably try selecting ALL, then deselecting the background, since the uniform color of the background will make this easier. To deselect, hold down Alt (on a PC --sorry, I don’t know the key for a Mac) while using a tool. If you’re using the magic wand, that will deselect all the white areas, for example. Then go for the gray areas in the background. You’ll want to play around with the settings in the tool options bar to use the wand effectively. 
Importantly, if ‘contiguous’ is not checked, then all areas of the color you click on will be (de)selected, whereas if contiguous is clicked, then only those on or adjacent to the point you click on will be chosen.
In this cupid example, if contiguous is not clicked, then alt-wand will also deselect areas of the wings, which is bad. You don’t want gaping holes in the wings.
Changing the Tolerance # in the options bar will modify how close to the target a color needs to be in order to be selected. A low number like 10 will select only pure white if you click on white. A higher number like 35 may select white as well as other light colors. Play with it.
Switch to the lasso for cleanup, perhaps. Play around with it until you’ve deselected all the backround. Then put that in a transparent gif as above. Result:
(I’ve been too lazy to clean up the edges, but you get the picture. Note that you can also use the eraser to clean up edges after pasting into the new doc, which is what I’d probably do here if I had time.)
Any questions?
