IRR stands for Internal Rate of Return. It's a way to measure the financial attractiveness of an investment over a given time period. It's typically expressed as an annual percent, e.g. 5%, similiarly to APR.
One way to understand IRR is to look at the sample cash flow table below:
In Investment A, $100 is spent at Year 0. Every year after that, $5 is received. At year 4, both the $5 and the original $100 is received. This will result in an IRR of exactly 5%. It actually doesn't matter if the original $100 is received at year 4, year 99, or year 1000. As long as $5 is received every year and the original $100 is returned at the end, the IRR will be 5%.
In Investment B, even though $5 is received every year, the $100 is never returned and predictably the IRR is terrible. However it will improve as long as the $5 payments keep coming in. The lower the payout every year, the longer it will take for the cumulative payouts to yield a positive IRR.
In Investment C, D, and E, the IRR are all 0%. As soon as the cumulative payouts equal the original invested, the IRR will reach 0%. That's because at that point, the investor would have made no profit, but also no losses either. IRR by default does not take into account external factors like inflation (hence the "Internal" part of IRR). However, the time value of profit is taken into account; the extra $10 profit for Investment F is received sooner than in Investment G, hence the higher IRR.
So as you might imagine, IRR is best used to compare the profitability of investments, not how soon they'd pay off (i.e. simple payback).
For a more technical explanation of IRR, including the role of discount rate, please see the Wikipedia article or refer to finance textbooks.