The term is an idiom; back-of-the-envelope calculations need not be performed on an envelope, or written on paper, or even written down at all. The defining factor is the use of simplified, scaled-down models.
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 Weisskof used back-of-the-envelope calculations to calculate the size of a hydrogen atom, a star, and a mountain, all using elementary physics.
A similar phrase is back of a napkin.
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