How to make background in image transparent?

Here is a solution using Gimp (2.0):

1. convert image to gif (assuming you want to use this for a webpage)
2. open image in gimp
3. right click on image and choose Layer -> Transparency -> Add Alpha Channel
4. right click on image and choose Select -> By Color, then left click somewhere on the background in the image that you want to make transparent
5. right click on image and choose Edit -> Clear
6. save image.

Online editor

Update: March 2015

Can do this online at http://www.online-image-editor.com (thanks to Marijan Beg for pointing this out):

It works both for images and animations. For transparent background: Upload image -> Wizards -> Transparency (diamond icon) -> click the colour background.

How to get rid of the (white) border around an image?

A very elegant solution is provided by the -trim option of ImageMagick's convert command:

convert input.png -trim output.png