The Stanford stereotype really varies. The ones I encounter the most are that students are known to be nerdy and wealthy. As a top-tier school, many self-proclaimed nerds do attend Stanford, but there's more to the students than just that. Ask anyone on campus to learn that they have a passion or talent that drives them, not just a high standardized test scores and good grades.
In terms of wealth, yes, many of my friends are from well off families. However, Stanford also offers a very generous financial aid packages and accepts applicants on a need-blind basis, which means Stanford doesn't consider your financial status when making application decisions. It's actually cheaper for me to attend Stanford than my own local public university back at home because of the financial aid I received!
To me, Stanford is a nerd meets jock meets hippy kind of environment. Our campus is very intellectual, active, and chill. Well, at least on the outside. There's something on campus known as the Stanford Duck Syndrome. Ducks look calm on the surface, but they're paddling furiously to stay afloat. Most stereotypes you hear about Stanford are probably true to some extent, because it's such a diverse campus full of passionate and driven people with many interests.