I have a blue Western Digital Passport external backup HD. Connects with USB and is no bigger than a smartphone.
Works flawlessly on my desktop, would be even better for a laptop because of its size. I have one the same size as my internal HD, which is actually overkill since it doesn't back up every single file (no system files, just personal files). Works silently and in the background every time your PC is on, automatically...you don't even know it's there but it's working. Doesn't slow the PC down at all. Has a cool utility to view what's backed up. They make them for Macs too.
I HIGHLY RECOMMEND this model.

(THE BLUE COLOR I HAVE)
The model I own:
My Passport Essential 500 GB USB 3.0/USB 2.0 Hard Drives ( WDBACY5000ABK )
More options:
Portable Overview
I got a 350MB on sale for under $40.