In the big picture, Notre Dame is a community of people who are serious about academics, athletics, school spirit and service (not necessarily in that order!).
Because the school is so selective, you will find that almost everyone graduated near the top of their class in high school. You will be surrounded by people who are smart, motivated and genuinely passionately about succeeding in college. Fortunately, this doesn't seem to have led to an uber-competitive atmosphere on campus--students are more than willing to help one another. The school itself is also extremely supportive of its students. Professors are generally very accessible, tutoring programs are all around, and the administration will intervene early to help students who are struggling. It may be very hard to get in to Notre Dame, but it's also hard to fail once you're there.
Athletics are obviously a big part of campus life, particulalry during football season. A home football game can consume an entire weekend's social activities (pep rallies, tailgating, game-watching, post-game partying, hanging out with visiting alums and family, etc., etc.). Most of the student body (although not me!) played sports in high school, and the intramural and interhall sports leagues are full of players who might have made the school's varsity program at a less athletically inclined college.
In terms of school spirit, Notre Dame is unparalled. The majority of students live on campus in dorms that fill the sorority/fraternity role, which makes them more bonded to and invested in the school. Many come from "legacy" families, with relatives who previously graduated, and have grown up loving the school. The alumni network is ENORMOUS, very active, and very inclined to take care of its own when it comes to job searches or other needs after graduation.
That passionate spirit for the university is equaled by a passionate spirit for service to the community. A very significant percentage of the student body is involved in service to some extent, ranging from volunteering at a local elementary school to building houses in Haiti. The university actively encourages and promotes these opportunities through academic programs aimed at service-oriented disciplines, dorm-sponsored service projects, and the sponsorship of lengthier service opportunities in South Bend and around the world.