I have created this image in Photoshop:
I would like to save this arrow to a .jpg file without the white background rectangle. How can I achieve that? I was trying to hide the background layer before saving, but this doesn't help.
jPeg images save automatically with a flat solid background. You will be able to achieve saving your image with a transparent background to .png format.
It won't work with JPEG format.
Some reference to assist transparent background
As mentioned, you can't make a JPG transparent. However, your reason for wanting it to be transparent is important for determining what format to use. For vector graphics, for example, EPS allows a clipping mask to mask out the background, and that format can used to place a graphic on a background in a layout program like InDesign, for example. That is why EPS is a preferred format for vector graphics to be used in print layouts. For other uses, the format, and the way the final image is composed, are both important.