I know of one way to do this, but there are other methods, but this is very easy. Do you have Paint.net? If not, download it. It's virus/spyware free, it's free, and it's a cross between the default Microsoft Paint and Photoshop, so it has many of Photoshop's features, such as effects and layers.
Open up your sprite in Paint.net. Where it says Tool, click the dropdown thing, and select the one that says "Magic Wand".
Two things to keep in mind: Flood mode and tolerance. Contiguous mode will only make a single area transparent, so any background not connected to your selection will not be made transparent. Global does a "total area" selection, meaning anything of the same color will be made transparent. Be careful with global, as if your sprite has any white in it, those will be made transparent. I recommend making any black/white in your sprites slightly off-color and not exactly Red=0/Green=0/Blue=0 (black) or Red=255/Green=255/Blue=255 (white).
Secondly, tolerance. This percentage determines how much of the image magic wand chooses to make transparent. Lower values means lesser amounts, and higher means higher amounts. 100% will make the ENTIRE IMAGE transparent. I would recommend 25%, but 15%-20% is probably better. Play around with this until you get it right, depending on how your sprite is. I use 15% usually, but I lower it if I have to work with colors that blend very finely to avoid losing them.
Now, just left-click your background and hit the delete key. Save it as png.