Full disclaimer: I'm a sophomore at Yale, my adviser last year was an admissions officer, and a friend of mine works in the admissions office. I'm also friends with the girl photographed, but that's irrelevant :)

This is roughly how admissions works at Yale:

1. An officer reviews your application for 20-25 minutes.
1a. Some kids are clear rejects. 1500/2400 SAT and a 2.7 GPA, with no activities usually does the trick. The director (or an experienced officer) reviews these.
1b. Some kids are clear admits. These are incredibly rare - something like 20 kids out of 26000 I think? The director also reviews these but they're pretty much auto admits - think multiple gold medalists at IMO, IOI, and IPhO, stuff like that.

2. The officer writes a summary sheet that contains what the officer feels about you, good points, etc, if they like you. If they don't, you get put in the "no" pile.
2b. I believe most of these "no" applicants are reviewed by another officer, but not as deeply. The goal is to find false negatives - most people in the "no" pile stay there.

3. The application goes to a committee of 3-4 officers. They all read your profile and debate about whether or not to let you in. Your admissions officer is supposed to argue in your favor, and the others can argue for you or against you. Usually, most candidates at this point are solid, so the officers argue about possible negatives.

Something like 70% of applicants are qualified to attend Yale, so it usually breaks down to what you'll contribute to the campus community, what kind of a person you are, etc. A lot of admissions officers were former Yalies, so I imagine they even ask "would I hang out with this person?".

4. If you're accepted, you're golden. If you don't make it past committee, you usually don't get in. Some people get waitlisted. They also review several of the "no's" to avoid false negatives, and some people might be brought to committee again. If you're borderline between "admit" and "reject," you will usually be rejected or waitlisted. There's just too many really good applicants.

There is no EXPLICIT comparison of Asians to Asians. Nobody looks at your application and says "Oh, another Asian, let me turn on my asian scale!" What happens, subconsciously, is that the stereotypical asian profile is "high scoring, high gpa, piano/violin, tennis, math/science."

So a lot of qualified asians get rejected because their admission officer can't find enough good arguments for them. Regardless of how qualified you are individually, Yale is trying to build a diverse class, so if you do the same thing as 1000 other candidates, it's very hard to vouch for you. "What do you bring to the campus that this other kid doesn't?
, and that's the end of it.

I don't think checking "Asian" or "not Asian" makes a huge difference, because in the end, it's your activities, recommendations, and essays that differentiate you. Once your scores are high enough, nobody is going to say "well Bob and Melinda are both cool, but Bob has a 2310 while Melinda has a 2270, so we should clearly go with Bob." That's just absurd - they always go with the person who will contribute more to Yale.

I hope that dispels some of the myths you see and hear. Correlation does not imply causation. People don't get rejected because they ARE Asian - it's usually the lack of differentiation. Again, I don't represent Yale or anything like that, and this is just what I've heard, but I believe it's fairly accurate, and for most Asians, they'll figure it out by your name, so a box isn't making a huge difference.

Edit: Legacy students (parent(s) went to Yale), recruited athletes, and under-represented minorities have higher admission rates than the overall pool. I don't know why or by how much or anything like that. This exists at almost every elite school.