I followed the instructions in How to use Preview for Basic Image Editing: Remove Backgrounds and Extract Shapes to use the Preview app to remove backgrounds with an image. The problem is that the "Instant Alpha" tool sometimes does not capture all the pixels and can't figure out how to use the rectangle tool to remove those remaining pixels. I tried highlighting the pixels and then pressing delete but it doesn't work. Can someone explain how to do it or recommend another free, lightweight Mac OS X tool to remove background to make it transparent? I do not want to use Photoshop or anything else expensive.
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.
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...
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.
You can use GIMP, which is free, but it runs in X11, which you will have to download separately. The GIMP has tools that are similar to Photoshop's tools.
You can Create Your Own Transparent Image Files With Preview
Basically using Instant Alpha button, select the area you want to make it transparent, then delete the selection, then it will be done.
Just see the link below.
It can be complicated. Go to Super PhotoCut in the App Store. It is really super to instantly remove background from photo within a few seconds. I used it a lot which saved me much time.