I use a program called GIF Construction Set from Alchemy Mindworks. As long as the picture is in GIF format it is just a couple of clicks to make the background of it transparent. This program also animates GIFs as well.
For the images that I draw myself I use Serif DrawPlus4 (not since I started with Windows 7) from Serif.com which has the option of saving the image as transparent (drawings on the mobile install page) also I can create animations with this program as well. I am not a great artist but I think I do fairly well.
Since my change to Windows 7, in early 2010, I have used the built in paint program.