Mrs. Ullman was actually a teacher at my seminary (Machon Raaya) and told us a lot about Binas, which she started the year after she taught us. candy613 pretty much got it down pat. I don’t know how much I would liken it to BJJ, though – Mrs. Ullman described it as a lot like Machon Raaya, with an added hashkafic dynamic. Small and more “in-the-box”/in-town than MR (but a lot less so than BJJ). The assignment candy613 described is actually modeled on Machon Raaya’s coursework, according to Mrs. Ullman.
As to their criteria for acceptance, I don’t think you have to fit the big-institution profile (father learning, blah blah blah) – like MR, she’s looking for a niche group that fits her hashkafic and itellectual ideas. So I’m guessing she’s at least a little selective, but not like BJJ and the like are (For instance: I got accepted into Machon Raaya very easily because my personality and work ethic fit their image for their student body, not because my father learns/works/wears white shirts/wears colored shirts/is well-off/etc).