Latin for "he undertook". To assume; to undertake; he has undertaken.
An extinct and archaic common law action for what would now be a simple claim for breach of contract and damages.
The assumpsit claim was that the defendant undertook to do something and then did not do it, or did it unacceptably and that the plaintiff was so harmed; thus, damages.
Assumpsit, like legal relics mandamus, prohibition and certiorari, were all the rage in the English law courts circa 1600 and featured a plethora of varieties such as express assumpsit (an undertaking made on the record) or implied assumpsit (presumed by conduct) and special assumpsit or general assumpsit.
The pictured case (below) is Gable against Forfester, 1661 and in case you can't read the fancy letters, where s's appear as f's, and capitalization is everywhere, here is what it says:
"Assumpsit on a promise to an infant, that is he paid money the Defendant wil make an insurance, that is not void, but voidable at election of the infant, as Dives (v) Manington case. If infant will permit J. S. to enjoy lands, J. S. will pay such a sum, this is good (and this motion made in arrest of judgment) but the very promise was to infant, if he forbear suing the Defendant, which is good cause of action be the promise void or true; and judgment to the Plaintiff."
See also trover.