As others have noted, the
O/D button is used to disable and enable the overdrive gear. What hasn't been discussed is why you might want to use this button, and in what circumstances.
The vehicle you have shown is a 9th generation Toyota Corolla. From the user's manual (which should always be your first source of information for your vehicle, specifically) :
Always drive your vehicle with the overdrive on for better fuel economy and quieter driving.
If the transmission repeatedly shifts up and down between third gear and overdrive when climbing a gentle slope, the overdrive should be turned off. Be sure to turn the switch on immediately afterward.`
Although steep grades exist that would create this condition, it would be more commonly encountered if towing a load. Disabling the overdrive when climing a steep grade ensures that the engine stays in third gear to provide climbing torque and will not hunt in and out of overdrive. If the grade is very steep the transmission will stay in 3rd gear naturally, but there will be some intermediate grades where it will want to problematically jump in and out of O/D.
This is particularly problematic in winter driving when traction is poor and the sudden torque increase from shifting down to 3rd gear up a slope (with the accelerator depressed) can cause wheelspin that can lead to loss of control. The city near where I grew up had an infamous hill like this and every winter there was always a handful of automatic transmission vehicles that would do this and go spinning off the road. Newer vehicles with traction control won't suffer as badly from this but it is an important thing to consider.
The other time you might want to disable overdrive is to take advantage of stronger engine braking when descending a steep grade. Going down a steep hill where your speed wants to run out of control you might be tempted to ride the brake, but this heats your brakes unnecessarily, causes excess wear, and can lead to momentary brake fade until they cool again.
Disabling overdrive forces the vehicle into the lower (third) gear and allows engine braking to help slow the vehicle, reducing the need to use the brakes on the way down. Again, always switch the overdrive back on when you reach the bottom of the slope.
Also note that (also from the 9th gen Corolla manual) :
Vehicles with cruise control — When the cruise control is being used, even if you downshift the transmission by pushing and releasing the overdrive switch, engine braking will not be enabled because the cruise control is not cancelled.
So if you want engine braking down a steep grade and you're under cruise control, first cancel cruise, then turn off overdrive and let the engine braking take you down the slope - you can turn the overdrive back on and resume cruise at the bottom. Newer vehicles won't necessarily have this limitation so, again, it is always best to consult your own vehicle's manual first.