What is the quickest Mac OS X app to use to remove the background of an image to make it transparent?
Solution 1:
You said "free," but you also mentioned "nothing expensive like Photoshop"; there are solutions to your problem that do cost something but are nowhere nearly as expensive as Photoshop.
Pixelmator is great for certain kinds of quick image editing. It's only $30 right now, though that will probably go up when the next major release finally comes out; I believe it's normally $60.
There's also Acorn, which is very similar and perhaps a hair's breadth easier to use, though since they've already had their first post-App Store major version release, it's more expensive at the moment ($50).
If it absolutely must be freeware, do have a look at Seashore, mentioned in a comment by Ian C.
Solution 2:
Just selecting an area and hitting delete just as you do with the Alpha tool should do it fine. I tend to get a good outline with the alpha tool, then zoom in and use the lasso selection tool to draw around any stray pixels - remember to complete the lasso yourself, or it may complete it for you and if the end is not near the start, it will just draw a big line between the 2 across whatever is in its way.
You don't say what OS you are using, but I've done it this way in every OS since Tiger...
Solution 3:
The best simple image editor is Seashore. It's an open-source Cocoa app built around GIMP's technology. That means it provides advanced photo editing capabilities for free and does it in a simple-to-install native app.