how can I make a logo backgroud transparent?

It is always best to create logos in Illustrator as then they are scaleable.

However.... if you really want to do this....  then duplicate the layer.  Then erase the background layer.  Then select the background (hoping it is a solid background) and then delete the background.  Next is to save it as a PNG like Silkrooster said.

Next time think about creating your logo in Adobe Illustrator..... as that is the best solution for creating logos in the first place.

To expand on Derek

Never save as a jpg because it does not support transparency.

If starting with a jpg, the background layer must be converted to a normal layer by dragging the padlock to the trashcan in the layers panel, because the background layer does not support transparency but a normal layer does. Then you can remove the background of the image and finally save as a png.

Assuming you have the Photoshop original and it has a transparent background, Save it as a PNG. if you don't have the original, say it's a JPG, you'll need to remove the backround to make it transparent before saving it as a PNG.