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1.
N. Old literary term for sexual copulation. The most famous being a quote from William Shakespeare's Othello where Iago informs the King that his daughter has been having sex with a black man. Though modern high school students generally assume that making the two back beast is a reference to doing it doggy style the literal meaning in the 15th century was sex face to face or in the missionary position.
"If you tell Momma I was spying on you, then I'll tell her you and Jorge were making the two backed beast."

William Shakespeare. Othello. (1605): Iago: ' I am one, sir, that comes to tell you, your daughter and the Moor are now making-the-beast-with-two-backs .'
by Keifermail July 27, 2008
 
2.
The "two-backed beast" is a euphamism for having sex. For some reason, people can't just say "having sex," they have to come up with a clever way to say it.
"Last night I was making the two-backed beast with my boyfriend"

Bumping uglies, knocking boots, burying the salami, etc
by Asty von Ferguson October 06, 2005