If an immigrant was rejected at Ellis Island (Rare, only 2% of immigrants were ultimately rejected), then they would be "shipped, steerage class and "tempest toss'd," at the shipping line's expense, back to Europe. So it was in the shipping companies favor to ensure they didn't bring any immigrants who were obviously mentally or physically disabled for fear of them being turned away. It also helped if you weren't a communist which could get you deported as well, a decision the Supreme Court ended up supporting 5-4 and still stands today.

I think others in this thread have answered this question admirably, but for further reading a great source is "American Passage: the History of Ellis Island" by Vincent J Cannato. I haven't finished it yet but it's very interesting and comprehensive. Covers Castle Garden era, too.

If I may ask a question, I would like to ask, "I just got to Ellis Island with my family, what now?".