Chill guys, ND definitely has a point in that frigid seemed overly eager in finding out about getting jobs at IBs with little knowledge and that IB is one of the more competitive industries out there, but then by the same token, this is an internet forum and you can ask people questions without fear of getting some doctored-up-PR-propoganda response which serves little purpose apart from glorifying the company/firm/bank.
Anyhow I feel like answering your original Q's (to the best of my knowledge);
- what do i-banks do?
Invest money. Their soul purpose is to make more money with the money they have. Sure normal banks do that too, but IBs tend to get their supply of funds from people interested in investing rather than people just saving up their money, and IBs invest in properties, commodities, businesses, shares, etc... to make a return whereas normal banks make money by charging interest on loans to other people/businesses. Maybe they should rename them to "Investment firms" to reduce the confusion.
- why are they so profitable?
Everyone has a speciality- theirs would be investing.
- why do their graduates earn so much?
Tough job, and as they say, "You have to sell your soul to work at Macquarie Bank."
- why do their graduates work so much?
Well, my cousins aren't exactly IB people, but they are forex dealers and commodity traders so they are constantly 'on the ball' keeping up with prices of various things from around the world at all times... I'd imagine that IBs are similar to that in the sense they need to always know whats going on, and as such, there's always things which can be done, calculations to be made, etc..
- how do i get myself a job in i-bank?
From what I've heard it's ultra competitive to get into any of the nice name IBs, so you have to do something which makes you better than the other candidates... apart from the degree, maybe extra curricular or work experience, or even personal experience might help you land a spot. The actual applications I think are done online like many other jobs nowadays.
(EDIT)
Maybe you might want to check out the wikipedia article-
http://en.wikipedia.org/wiki/Investment_bank
... btw whilst I may be in the Finance industry, I don't work in any of the IBs nor have I studied Finance.