Copy and paste the selection onto a transparent canvas.
Larry Berman
And save to a format that allows transparency, such as gif.
I assume you have just the one ["background"] layer. You need to convert that to a "normal" layer by double-clicking on its name in the Layers palette. Then, deleting will leave transparency instead of the background colour [white by default]. Of course, you need to save in a format that supports transparency – as suggested, GIF is a good bet.
Newsport,
If you are making a transparent fie for the web you can just use "Save For Web", Select GIF as the file type and map the colors that you want to be transparent. Just remember that GIF files are limited to 1 Bit transparency and 256 colors so they are mostly used online.
If the logo is for video or print you will probably want save to a TIF or PSD depending on the application you will be using the file in.