I get the impression from TV, movies, etc, that Americans have to pay all their yearly income tax in one lump around the 15th of April? Is that true?
No. Money is withheld all year from taxes. This is either done automatically (if you work for a company) or if you are self-employed you send in quarterly estimated withholding. April 15 is when you figure out exactly what i owed and wither send the balance or get back a rebate. There is a penalty for under witholding (sp?).
Now that I write this down, it seems complex.
That’s when you are responsible for having it reported and paid if needed. I pay an estimated fraction out of each paycheck so come April 15th, I’ll hopefully have my obligation met. If the amount taken out each week isn’t enough come April 15th, I’m responsible for making up the difference. If more has been taken out than I’m responsible for, I get a refund (with no interest paid on the amount). You can apply for an extension, but you still have to pony up by August IIRC.
Oops; withheld from income.
That’s probably the understatement of the year. Nothing about income tax is simple. I’m pretty sure that’s legislated into the whole process. 
Ok - pretty similar to Oz, then.
As was mentioned, most people have income tax (as well as Medicare and Social Security Tax) taken out of every check. You determine how much income is taken out by how big your family is (the people you are caring for)- so that number varies. Medicare and SS are 7.65% and .9%-- I can’t remember which is which, though.
If you don’t pay out of every check, you are supposed to pay quarterly estimated payments (January 15th, April 15th, July 15th, Oct 15th) and there are a few ways to calculate that amount. I can explain if you want, but I promise that it is nothing exciting.
Come April 15th, you prepare your tax return. If you paid more throughout the year then you are liable for, you get a refund check. If you didn’t pay enough, you owe them some money. I have a friend who got a refund of $1 (I laughed for days about that one) and another friend who owed $5.
Nitpick:
Jan 15, April 15, Jun 15, Sept 15 are the quarterly deadlines.
For those in other countries, you can file your income tax return anytime after the first of January, April 15th is the deadline. So if you’ve a refund coming, you’ll generally file as soon as you can, but if you owe, you wait until the deadline.
Satch
Unless, like 99% of Americans, your piddling tax refund is less valuable than time spent procrastinating. 
Similar rules for Canada (tax deducted from each paycheque, pay tax by quarterly installments if you are self-employed, being able to send in the return early if you wish, etc. etc.) except we get to send in our tax returns two weeks later than the folks south of the border (April 30).