Short and sweet:
IRR, ROI and NPV all recognize the time value of money - a dollar received next year (for example, as savings from reduced errors) is worth less than a dollar today.
NPV is the net present value (the value today) of all those future savings - simplistically, $1 today, +$1*0.9 for next year's savings (at a 10% interest rate), continuing for all the years; and by the way applied as well to any investments made. Lots of different ways to do this calculation - see a finance book.
ROI is the rate of return on your investment-
again simplistically, if you invest a dollar now and in a year you have $1.10, you've made 10% on your money. Again, lots of variations on ways to do this.
IRR (internal rate of return) is often used within corporations as a "hurdle rate" - for example, "we will only make investments that have an IRR of 13.5%". Used in this context, the ROI for a project is compared to the required IRR, to see if the project is acceptable.
Payback ignores the time value of money - if you invest $10,000 in a training program, and save $4000 per year from that effort, your payback is 10,000/4,000 = 2.5 years.
A finance book will give you more details on mechanics, but this should help you find what you are looking for.