old row or new row?

First, originally "Old Row" referred to fraternities on the west side of campus that did not move to the east side when new fraternity row opened in the early 1960s. It did NOT refer to sororities.

Back in the 1980s, a beer company partnered with a bunch of fraternity guys for a loose organization called "Old Row." It was a one-time party, funding by donations from the fraternities, with copious beer from the involved company.
In the spirit of equality, eventually sororities were invited to join "Old Row" and throw in their money, too. Supposedly you have to be invited, but several sororities' governing bodies forbid them from spending money on alcohol, so several aren't members because of that. It's really not that exclusive, and it has nothing to do with the status of the sorority. Uually a sorority will not get "kicked out" - they'll decide they don't want to spend the money on the party.

As for joining...There are about 6 to 8 sororities that are the most "in demand" and supposedly "exclusive" - AXO, XO, AGD, DDD, KD, KKG, PM - and that will vary somewhat depending on who you talk to and what kind of beef they have with a particular group.
All of them have various requirements for membership, the foremost of which is grades. Other requirements (which may or may not be common knowledge) may include involvement in extracurriculars, volunteer service, and honors and offices. And if you know current members, that will help you with any sorority.