The question of whether a school’s designation as "elite’ should be a relevant consideration in choosing a school is a separate question. On that, I think the answer is no, in and of itself. But I do think schools that are considered elite generally have a lot of other positive qualities that are relevant to choosing a school.
On the question of what makes a school “elite”, at least on CC I think it’s largely a function of prestige and reputation, which is also closely correlated with ranking and selectivity. There’s no universal definition, and the cutoff line will vary by individual.
I think there are only a handful of schools that would be almost universally considered elite: Harvard, Yale, Princeton, Stanford, MIT, CalTech, Chicago, Columbia. My guess is that most would also include the other Ivy League schools, a few top LACs like Amherst, Williams and Swarthmore, and a handful more like Duke and Berkeley.
Beyond that, you’re likely to get some debate. Is Notre Dame elite? Vanderbilt? Rice? Wellesley or Pomona? I expect some would argue yes and some would argue no. At the end of the day, does it really matter? They’re all excellent schools (as are many others) where you can get a great education.