A back-of-the-envelope calculation is a rough calculation to test a hypothesis. It is not necessarily written on the back of an envelope. It is more trustworthy than a guess, but less definite than a mathematical proof.
The idiom originates in the practice of quickly jotting down calculations on the nearest available piece of paper, such as an envelope. The defining characteristic of back-of-the-envelope calculations is the use of simplified, scaled-down models. A similar phrase is "back of a napkin".
In the hard sciences, back-of-the-envelope calculation is often associated with physicist Enrico Fermi, who was well known for emphasizing ways that complex scientific equations could be approximated within an order of magnitude using simple calculations. He went on to develop a series of sample calculations which are called "Fermi Questions" or "Back-of-the-Envelope Calculations" and used to solve Fermi problems.
Fermi was known for getting quick and accurate answers to problems which would stump other people. The most famous instance came during the first atomic bomb test in New Mexico on July 16, 1945. As the blast wave reached him, Fermi dropped bits of paper. By measuring the distance they were blown, he could compare to a previously computed table and thus estimate the bomb energy yield. He estimated 10 kilotons of TNT; the measured result was 18.6.
Another example is Victor Weisskopf's pamphlet Modern Physics from an Elementary Point of View. In these notes Weisskopf used back-of-the-envelope calculations to calculate the size of a hydrogen atom, a star, and a mountain, all using elementary physics.