If the background was already transparent, do this:
Go to File > Save for web (Alt + Ctrl + Shift + S) and save it as a PNG or GIF, which support transparency.
If the background wasn't already transparent, do this:
First, create a new layer (Ctrl + Shift + N). In the layers palette, click on the background layer, and Ctrl + A to select the whole thing, go back to your new layer, and Ctrl + V it into that one. This layer is for if you need to start again, so click the visibility icon (little eye) next to your new layer.
Now, switch back to your background layer, and press Alt + Ctrl + X to get into the extract filter. A screen will pop up, and you'll see that you're using a highlighter tool (Note: Zoom in with Ctrl + Plus). Now, you need to use the highlighter tool to highlight only the edges of the image that you want to keep.
Here's a picture of what to do. I use a highlighter of size 3 to highlight the edges.
<img src="http://img.photobucket.com/albums/v92/ifiredanuke/extract.png" />
After that, click Extract, and you should have your image minus the background. Then just Ctrl + Alt + Shift + S and save as a PNG or a GIF. :p
Here's one I prepared earlier, it's not the best, but I extracted it, and applied a bit of blur and lowered the opacity of the loopy bits slightly.
<img src="http://img.photobucket.com/albums/v92/ifiredanuke/apcavatar.png" />