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What happens to offshore money when it is brought onshore?

Depends whether you bring it back in during an offshore amnesty with reduced tax rates (the US is due one soon), but generally you want to avoid bringing it in at all. But how can you use it if you don't bring it in?

Well, most big corporations (e.g. Apple/etc) don't need to bring it back in (and pay tax on it), they use the money in offshore accounts as the guarantee for borrowing the money they need in the US (at a nice low interest rate). That way they get access to the equivalent sum of money without having to pay the taxes to bring the original dosh back into the country. The interest payable on these loans is also tax deductible (on company profits) to some degree too so it's a double win.