Rutgers, the State University of New Jersey was founded in 1766 as Queen's College also by Royal Charter, but by the grandson of King George II, George III (also known as "that dude who believed in taxation of Americans without representation and later made a habit of talking to trees"). It was renamed in 1825 after Colonel Henry Rutgers, a Revolutionary War officer and well-known philanthropist in the hope of buttering him up for a donation. He ended up donating a $5,000 bond (and a bell) and apparently gave the impression that he would donate much more (he didn't).
Today, Rutgers is the official State University of New Jersey, a designation that encroached upon it with the awarding of a land grant in 1864 of 210,000 acres in, of all places, Utah (amidst feverish competition with Princeton, no less). By 1945, Rutgers had officially been designated a state university. It also gobbled up the University of Newark (now Rutgers-Newark) in 1946 and the College of South Jersey (now Rutgers-Camden) in 1950. Both schools arguably became worse off after the merger. Presently, Rutgers is thinking about nibbling away at UMDNJ and NJIT.
Relation to the Ivy League
Rutgers sometimes tries to get cute by claiming it was "invited to join the Ivy League". Which is preposterous, as the Ivy League (an organisation open only to private schools) was not founded until 1954, well after Rutgers was designated a public university. The claim possibly stems from the fact that in 1793, a proposal to merge Rutgers and Princeton, as both colleges were running short of both funds and enrollments, failed by one vote - cast by a Rutgers Trustee, as it were. This is also problematic for the reason that not only did the Ivy League not exist back then, but Princeton wasn't even called Princeton (it was the "College of New Jersey").
Rutgers' relationship with Princeton is roughly analogous to Cornell's relationship with Columbia.
Relation to Columbia
Rutgers also tries to claim that it is a "sister school" of Columbia, since its original namesake was "Queen's" rather than "King's". However, the "Queen" for whom Rutgers is allegedly named is Charlotte of Mecklenburg-Strelitz, who was not George III's sister, but rather his consort. Ergo, a more proper designation for Rutgers would be "Columbia's bitch", a fact that will no doubt spurt an obligatory Barnard joke. Perhaps appropriately enough, 5/11 members of Rutgers's Board of Governors are Columbia alumni, while only 3 are Rutgers alumni. To make it even more absurd, the person for whom Rutgers is named, Henry Rutgers, was a member of the King's College Class of 1766.
Nonetheless, ties between Columbia and Rutgers were, at one point in the past, somewhat strong. An entire class of Columbia juniors in the early 19th century threatened to withdraw and petition for admission as a class to Rutgers when the Trustees were considering expelling a particularly troublesome member. And supposedly, there were once "annual Queen's College-King's College (Columbia) debates", but no one has heard of one in recent memory.