All aircraft used by airlines have what is called an MEL, a Minimum Equipment List. The MEL list all the stuff that is allowed to be inoperative on the aircraft during normal operations. It also list required proceedures that are to be used to compensate for the broken equipment.
A simple example might be a 'wing inspection light' that shines down the leading edge of the wing, allowing the flight crew to see if any ice has formed. If you notice that this light is burned out, you would look in the MEL and find that it is ok to fly, as long as you stay out of icing conditions at night.
The flap degree markings on the wings are probably for a similar purpose. If the indicator in the cockpit is broken, it would be possible to complete a flight if other means are used to verify the flap position. It may also be used during preflight to make sure that the flaps are in the correct postion.
Some jets with 'T' tails that use a jack screw to move the entire horizontal stabilizer will have markings to show when the trim is set correctly for takeoff. This will allow an aircraft with an inoperative trim indicator to complete a flight, by having the flight crew visually verify that the trim is correctly set for takeoff.
If you are curious about this sort of thing, you can find a list of all of the available Master Minimum Equipment List (MMEL) on the FAA website. Just use the search function. A MMEL list everything that can be inoperative on a particular aircraft but has to be customized by each operator depending on what regulations they operate under and how their aircraft are equipped. You will be amazed how much stuff can be broken on an 'airworthy' aircraft. The PIC has to make the final call, and it is possible to be in a position where the aircraft is legal to fly and have the Captain refuse the flight under certain circumstances.