Where are you going to use your logo?
JPEG does not support transparency. Usually you need to have the logo in PSD format and then add the background color you need.
To remove the white from your logo try the magic wand tool, click on the background. It will select all the white pixels.
Then, invert your selection. Look in the menu Select/invert.
In your Layer window double click your Background Layer. It will convert it in a layer (the name will not be in italics anymore).
At the bottom of the Layer window there is an icon for creating a new mask. Click on it and your selection should be converted into a mask, letting the background transparent for your logo. The transparent background in photoshop is represented with little grey squares.
Save it as a PSD file.
If you need it, for example, with a yellow background just create a new layer, put it behind your logo, fill it with yellow and save as your document as JPEG or TIFF. Remember add a descriptive name as: LogoBgYellow.jpg
Some software accept PSD files with transparency. Also, if you are working for the web, GIF files can be exported with transparent pixels.