Here's how it works electrically:
Voltage is provided through a "hot" wire from the ignition exciter coil, when the engine is running, to the bulb. The power goes through the bulb and seeks ground through the oil pressure switch. So, you have a power wire (typically yellow) to the bulb and you have a ground wire (usually black) from bulb to switch. When the engine oil pressure rises after start up, the switch closes and grounds the bulb.
This is a RUDIMENTARY warning system for low oil pressure. It is IMPORTANT that you verify oil pressure to prevent engine damage. It is always a good idea to put a mechanical gauge in the oil pressure port although I don't know anyone that actually does. I am adding this information as a CAVEAT. I know of no published specification for MINIMUM oil pressure but I like to see at least 15 psi at 2000 rpm. There! I warned ya!
Those lights do burn out. A quick check would be to find the oil pressure switch on the starboard side of the engine behind the ignition coil and slide the protective boot from the electrical connector. Using a voltmeter, run the engine and check for voltage present at switch. If there is no voltage at the switch, it could be that the bulb is burned out. To check, unplug the the bulb's ground wire and do a voltage check there. If you have power, then bulb is ok and there is a wiring fault between the bulb ground connector and the oil pressure switch.
If you do not have power, then unplug the bulb's "hot" wire (should be yellow in color) and check for power. If you now have power, the bulb is definitely bad. You MUST use caution when doing this testing to NOT allow the "hot" wire to touch ground at any time. Otherwise, you can burn out your exciter coil. If you don't have power at the "hot" wire, there is likely a problem with the exciter coil under the flywheel. Although, there is a connector for that wire that can have high resistance. I have not seen that connector give problems.... but you never know.
If you initially found that you have voltage at the oil pressure switch, then you can dispense with all the above and concentrate on the switch itself. First, use a "jumper" wire to the switch terminal and attach it to a reliable ground on the engine. If the bulb now illuminates when the engine is running, you will either need to remove the switch and replace it or you can try cleaning it out and re-installing it to see if it works. The switches on these get "sludge" built up in them and, sometimes, all it takes to get them to work again is to spray some contact cleaner up into the inlet and the let it drain out and dry up.
If you find that cleaning the switch and re-installing OR replacing it with a new one still leaves you with no oil light, it is HIGHLY likely that you have VERY low oil pressure and continuing to run the outboard like that will eventually lead to catastrophic engine failure.
That's all I got. I hope didn't confuse you.......too much...
....Good luck.