There are many ways to remove the white background.
You might try using the magic wand tool to select the white, press delete to remove the white background then use File>Save for Web.
If there is still some of the white remaining around the logo, either use a layer mask from your selection and clean up the edges or try
Layer>Matting>Defringe and then save the file.
You might use png-24 in the save for web if you want quality over file size.