Here is how someone who says he interviews for Harvard says the process works:

[T]he AO (admissions office) offers to schedule interviews for every candidate that applies, wherever there are in the world. Cities like NYC have a surplus of alumni and often have two year waiting periods just to be allowed to interview. In the Bay Area, we are generally a bit short of interviewers and will focus on candidates ranked 3 or higher by the AO. The goal of the interview is NOT to get facts (e.g. SAT scores, list of accomplishments, rankings). Those details will all come out in the application itself. Instead, we try to get a sense of the candidate as a person and most importantly how they might fit in with their potential peers in college. The AO asks interviewers to report on "What is it like to sit and talk to the student? What are the studentís motivations or aspirations? Does the conversation flow freely? What kind of roommate will this student be?" The last one is my favorite and usually what I like to focus on.

In the end, the interviewer will write up a report with their impressions of the candidate along with specific examples and quotations from them. Strong candidates will get longer write ups. Non-competitive candidates will get shorter ones but any reservations we have should be explained. We rank students in several areas from 1+ (the highest) to 5-. All students start with 3 and move up/down from there. For example, in extracurriculars, a students who participates actively (5-10+ hours/week plus competitions) in Math Olympiad (a popular club here) will get a 3. If they have won some sort of state or regional recognition, they will get a score in the 2 range (+/-). If they make it to the national or international level, they may get a 1 (extremely rare). Most accepted students will be in the 2/2+ range. During the final round of admissions, the whole admit committee at Harvard will put the interview report on a projector while a regional "advocate" reads through the highlights of a candidate's application before a decision is made. Since I don't work on the admissions side, I can't say for certain how much weight the interview has but a good report will certainly nudge up candidates who are already competitive on paper.

As we know, getting in is very difficult:

As far as college interviewing goes, DW and I have being volunteering with the local Silicon Valley Harvard club for the past 6 years. In that time, we have interviewed over 30 candidates. Of those, 3 were wait-listed and only 1 was accepted (admit rate for 2015 was 5.3%). Many, many more were, by any standard, truly exceptional students (near-prefect scores on their exams, 4.0+ GPAs, 8-13 AP courses, diverse extracurriculars) but competition is quite fierce in the public school districts here. Many of the students have highly educated parents who work in tech and have similarly high expectations for their kids. We love meeting the students at the various recruiting events but it's the parents who give us the most trouble!