How can I take an image with a white background and make it transparent

The GIMP will allow you to convert a single colour to alpha, making it transparent.

The basic flow requires right-clicking on the image and choosing:

Filter > Color > Color to Alpha

And then selecting the white background. The GIMP will replace all white pixels in the image with transparent pixels.

Here's a good tutorial showing you how.

you can use Photofiltre (small, fast, free and portable):

1. load the image

2. go to Image > Transparent Color...

3. choose the color (white), set the desired tolerance with the slider, click OK

4. go to File > Save as... and save the image as .PNG

I haven't seen a solution using Paint.NET here yet. It's an awesome lightweight image editor. To quote Jeff: "It raises the quality bar,"

1. Download, install and run Paint.NET
2. File > Open > Select your image

3. Click the Magic Wand tool:

   

4. Click the white background on your image
5. Press Delete
6. File > Save

You can also select the tolerance of the magic wand tool before your use it to get rid of the "crust" or whatever you want to call it around your leftover image.