Hi Stargazer9,
The school's rounds are how they structure their admissions process. You can only submit an app to one round at any given school. The school fills their upcoming class with applicants they accept in each round. Generally, the earlier round you submit in, the "easier" it is to get in, because there are more slots open. Each round is a discrete unit, and the school's admissions process goes through the same cycle within each round, then repeats again for the next round. On their side, that cycle is reviewing and assessing applications, conducting interviews, and deciding: accept, reject, or waitlist.
Most top schools in the US have three rounds: R1 in October, R2 in early January, and R3 in March or even April.
To make it even more confusing, some schools have more than three rounds, and one school (MIT) has just two rounds! And, even worse, some schools have "rolling" admissions, or "Early Action" rounds (Duke, Tuck, and Columbia have this; with Columbia, it's binding, meaning if they accept you during their Early Decision round, you are already committing to them that you'll accept their offer and attend). There's all these different variations and you'll need to study the way your target schools do it.
Here's an example of how it works at UCLA this season:
The Round 1 deadline is Oct. 20, 2010. That means you have to have your essays done, your letters of recommendation submitted, and the entire online application filled out with your resume uploaded, and click "submit" by that day. There *is* an advantage to applying early to UCLA, so get it in before the deadline if possible. And, very important, you need to have your GMAT taken by Sept. 30 (this is a nuance for UCLA, for other schools, you just have to have the score available on the day you submit). Then, the waiting game begins! That's the hard part. The adcom reviews your app, and if they like it, they'll invite you to interview. Interviews are done by a student or admissions staff person, and they can be on campus or by an alum in the city where you live. UCLA says they're releasing R1 decisions on Jan. 12, 2011. This means that you won't hear back before the Round 2 deadlines at many other schools, unfortunately. With schools like Columbia or Duke, if you apply in their "early" rounds, then you'll definitely hear back with a "yes" or "no" in enough time to put together a Round 2 app for another school that would be perhaps further down your list.
You said that you're taking the GMAT at the end of this year, which means you could only apply to Round 2 or later rounds. If possibly, you should re-think this, and take the GMAT sooner. It's best to submit in R1 for the schools at the top of your list, and for the most competitive schools. A R2 application still has a very good chance, too, but once R3 rolls around, next spring, most schools will have most of their entering class filled up, so there will be fewer slots available, and it's harder to get in. However, you should never rush it! It's always better to submit a strong application in a later round than to cut corners or do a less-than-perfect job to submit in an earlier round. You should have plenty of time to get ready for R1 from here.
EssaySnark focused on the "rounds" question but it looks like you have a lot to work with in your profile, so Good luck with it!
ETA: If you want to go into healthcare, consider Vanderbilt - they have a very very strong healthcare program. UNC-Kenan Flagler might be an option to consider as well.