Surface element doesn't work
When an electric element doesn't work, either it isn't receiving electric power or it's likely that the element or receptacle it plugs into is faulty.
Check the other elements. If none work, the circuit breaker has probably tripped. Reset it at the electric panel. If this reoccurs, call an appliance repairperson.
If just one burner isn't working, try to pinpoint the source of the problem-- likely a bad element/burner or a problem with its connection receptacle. In some cases it's the switch or the wiring. To replace a bad switch or repair the wiring or replace the receptacle, call an appliance repairperson.
A burner that plugs into a receptacle is easy to test. After turning off the range's power at the main electrical panel, just unplug the non-heating burner, plug it into another working receptacle, restore the power to the range, and test it. If the burner works, you know its original receptacle is probably faulty. If it doesn't work in the good receptacle, the burner is bad and should be replaced. Buy a replacement and simply plug it into the receptacle.
When you remove the burner, look for burned wires or a charred receptacle. Check the receptacle or terminal block to see if it's cracked, loose or looks burned. If you notice any of the above signs, replace the faulty component, too. If the male burner prongs are corroded, always change the female receptacle it was plugged into.
A flip-up burner requires a little more work. After turning off the power, tilt up the burner, unscrew the small screw that holds it, and then slide it out. Disassemble the insulator block by prying off the clips. Then unscrew the wires from the element, replace with a new burner, and reassemble.
To test a burner with a multi-meter: First remove the burner from its receptacle, as discussed above. Set the multi-meter to the Rx100 setting (or, for a digital meter, to [omega symbol] or k[omega symbol]. With the red lead connected to the positive jack and the black lead to the negative jack, touch the black probe on one of the heating element's terminals and touch the red probe to the other terminal. The needle should jump from the infinity reading to the right, indicating a properly "closed" circuit. If it doesn't, try the probes on a different part of the metal contacts. Still no reading? Touch the two leads together to make sure the meter is working (the needle should jump). If it works, there is an internal break in the element and it will need to be replaced.
When you're finished, turn the power back on.