Let's see if I can get my high school maths right.
With simple interest, the formula is simple.
If you borrow $100, at 6% per annum simple interest for half of a year, you must pay back:
$100 * (6/100 * 1/2 year) = $3 interest, plus the original $100 principal.
But, if you borrow with compound interest, then you have to pay interest on the unpaid interest. The rate of compounding (how often you calculate the interest) makes a difference.
If it compounds every 6 months, it is still the same amount: $3.
If it compounds ever 3 months - i.e. twice - it is:
$100 * ((1 + (6/100 * 1/2 year)/2)2 - 1) = $3.022 interest, plus the original $100 principal.
If it compounds every day - i.e. 183 times - it is:
$100 * ((1 + (6/100 * 1/2)/183)183 - 1) = $3.045 interest, plus the original $100 principal.
So the frequency of the compounding is important, even though the nominal interest rate (6% per annum) didn't change. The nominal rate is also known as the Nominal Annual Percentage Rate (APR).
Governments (and most legitimate businesses) understand that the nominal APR is misleading, and they insist that people publish the Representative APR (a.k.a. Effective APR, EAR, Annualised APR, or AAPR) which is standardised as being the interest rate you'd pay if you borrowed for one year.
Now, lets use Wonga's figures, to see how the representative figures look:
If you borrow $100 for 1 year at a nominal rate of interest of 360% per annum compounding every day (non-leap year), you actually pay:
$100 * ((1 + (360/100)/365)365 - 1) = $3495.84 interest, plus the original $100.
This corresponds to an Representative APR of 3495%, but there may be other charges that make the real figure even higher.
Now, Wonga is right that, because the loan is much shorter than a year, your total bill will not come to that figure. However, it is a representative figure because it allows you to compare their interest rates to other lenders.
Wonga can honestly quote 360% (or 1% per day) as the Nominal APR, but (as they admit) the regulations require them to also state the Representative APR which is much higher.
It appears that they charge simple interest over the short period of the loan (plus a fee), which I assume means the Representative APR represents the worst case of a 1 day loan. If you borrow for 30 days, the annualised APR would be lower.
Here's one online tool to try this stuff out for yourself. It computes the last figure as $3522.43 - I am hoping the difference is taking into account leap years and rounding errors.