The American university evolved from the British institutions like Oxford and
Cambridge. Harvard was the first university in North America having been
founded in 1636 (interestingly, Santo Domingo, Dominican Republic 1538,
Peru and Mexico 1551, and Cordoba, Argentina 1613 are older. If you are into this
kind of stuff, the madrassa Al-Karaouine en Fes, Morocco is in continuous operation
since AD 859. Bologna (1088), Paris (1091), Oxford (1096) are the oldest in Europe).
All this is nice, but largely irrelevant. Those institutions have very little to do
with the modern university. Universities in the US took an important acceleration
in 1862 with the Morrill Land-Grant act. This was a law passed by Congress
(during the Civil War!) in which gave 30,000 acres of federal land to each state
per each member of Congress the state had. Proceeds from the sale of these
lands were used to fund universities by the states. Most of the universities so
founded are the current large “state universities” (LSU, Southern are examples). A
notable exception is Cornell, which is private.
U. of Arizona, U. of Alaska, Colorado State, U. of Connecticut, Delaware State,
U. of Maryland, Missisipi State, Rutgers, NCSU, Ohio State, PennState, Clemson,
You may also hear sea-grant, space-grant (LSU is both) and sun-grant universities.
Those are separate more recent programs that do not really involve a grant.