Hi. The following assumes that you have an automatic transmission -- it's use on a manual transmission is different.
The "overdrive" button is really a misnomer, as it perhaps should be called the "no overdrive" button -- but more on that below.'
The transmission *normally* incorporates all four of its forward gears (the top-most being an overdrive, but for simplicity let's just think of it as your highest gear, 4th, or the one which combines fastest road speed with lowest rpm's).
Also, notice that along your gear shift gate, the forward gears are marked 1, 2, and D (a.k.a., "4th"). There's no indication of 3 -- and that's because the little OD button controls the shifts between 3 and 4, rather than the big shift lever (again, discussed below). These choices, 1, 2, and D, really indicate how high up the tranny will shift. E.g., if you're in 1, the tranny stays in first (it won't shift up any higher); if you're in 2, it will start in 1 and shift only up to 2; and if you're in D, it will normally start in 1 and shift through 2 and 3, eventually into 4.
However, if you push the button (and the dash light comes on), then the tranny will only shift up as high as 3rd (i.e., you've locked-out 4th, the overdrive gear). Or, if you in 4th and you push the button, the tranny will shift down into 3rd. Got it?
Now, when do you use it? Well, first of all, you keep the tranny locked out of 4th whenever you tow anything above 1,000 lbs. That means the whole trip! It's to ease the strain on the engine and transmission.
And you can freely use the button to stay in (or force a downshift to) 3rd gear whenever you want more torque or acceleration (speeding up, going up a long grade, etc.), or more holding back (going down a long steep grade). Remember that 3rd gear will force the engine to run about 1,000 or a bit more (depending on your speed -- it's a proportional increase) faster than in 4th.

Hope this helped.