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.
You can't save transparency to a .jpg. The file format simply doesn't support it. Anything that is transparent will become white when saved to a .jpeg.
Try .png or .gif, those file formats do accept transparency.
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.
Hide the background layer first. Then try "Ctrl+ Shift + Alt + S" to save for web and choose png. Or just save as and choose png as your filetype.
Opt for PNG32 which allows for full alpha transparency. That's what I use.