A no-background favicon for my website?

You want a transparent image. You'll have to create a PNG file with transparency. JPEG doesn't support transparency. Steps may vary depending on the software you use. But basically you have to work over an image with alpha channel (transparency). You have to select some parts of your image and remove them. In its place you'll see some kind of pattern representing the background.

Later you have to specify transparency option during PNG export.

UPDATE

Using GIMP create an image and add an alpha channel. Draw as desired and be sure you have some transparent area like explained above. Then choose File > Export... and select a destination file with PNG extension. GIMP displays an export options dialog like this one:

Be sure you have "Save color values for transparent pixels" and "Save background color" options selected, then press Export button.

I highly recommend you using a Favicon generator. Like this or this, after which you'll want to set this code in your header in order for the icon to show up as a favicon.

<link rel="shortcut icon" href="/favicon.ico" type="image/x-icon">
<link rel="icon" href="/favicon.ico" type="image/x-icon">