In the simplest possible sense, a backup is another copy of some piece of data. In the physical world, you might photocopy a piece of paper; on your computer, you do the equivalent by making a copy of a file. The great thing about digital copies is that, unlike photocopies, they're absolutely identical to the originals. A lost or damaged original would therefore be of no consequence as long as you have a backup.
Continuing with the photocopy analogy, suppose you've made a copy of a will, a contract, or some other important legal document. You probably wouldn't keep the photocopy with the original because if your office were burglarized or the building burned down, the original and the copy would both be gone. Common sense dictates that the copy — the backup — be stored somewhere else.
The same is true when backing up digital files. If I duplicate a document and store the duplicate on the same hard disk, I have a tiny bit of protection; if the original file were somehow damaged or deleted, I could use the copy. However, anything that affected the entire disk would also affect my backup. Storing the backup on another drive inside the same computer is a notch safer; storing it on an external drive is safer still; and storing a copy in an entirely different location is best of all. Better yet, I could maintain more than one backup copy — for example, one that I keep near my computer (for convenience) and another I store in a separate location (for safety).
In a ...