I've always been perplexed about that thing that americans call "dating".
Actually, from my perspective "dating" is sort of a concept that exists and can be discussed by never actually happens. People start hanging out, then they start "going out" as booksandlibretti described (perfectly, I might add) and then there's boyfriend girlfriend. If someone says they're "dating someone" that's the same as "going out" with someone or are their boyfriend/girlfriend.
If someone says they're "dating" as opposed to "dating so-and-so" that means hanging out with a lot of guys and (maybe) going on dates with them. Men are never "dating" really. "A date" would be much more formal then just 'hanging out'.
Most the term is a hold-over from times when people were expected to 'court' each other, and seek out relationships in a much more formal way.