An electric range's element is basically just a resistance wire suspended inside of a hard metal alloy bent into various shapes, separated from it by insulation. When power is applied to it, the resistance wire generates heat which is conducted to the element's outer sheath where it can be absorbed by the cooking utensil or the air inside the oven cavity.
When this type of element fails, the internal resistance wire breaks causing an open circuit. Since the circuit is now open, no electrical current can flow through the element to generate heat.
The insulation between the resistance wire and the outer coating can also deteriorate. When this occurs the inner resistance wire can come into contact with the outer sheath causing a short to ground. The force of such a short can blow a hole right through the outer coating, melting it and sometimes damaging cookware being used at the time.
Oven elements have occasionally been known to light up like a fireworks' sparkler when their wire filament shorts to ground. This effect may continue even after power has been disconnected from the range. In all cases there is no actual possibility of fire as once the element material has been fully exhausted by the effect, it will eventually go out on its own. It can create quite a disconcerting light show for anyone nearby at the time though! Power should be totally disconnected from the appliance in such a case until it can be repaired.
A short of bake or broil element can also sometimes damage components of the control circuit (thermostat, selector switch, etc.) as well. On models with electronic controls the control itself is very susceptible and may be damaged by such a failure. If the element doesn't work after being replaced and full power restored to the appliance, components of the control circuit may need to be investigated.
To test an element for continuity the appliance should first be disconnected from power. After the appliance has been made safe to work on, the element needs to be isolated from the rest of the electrical circuit by removing at least one of the connecting wires. Once that is done, an ohm meter or continuity tester's leads can be held against each terminal of the element.
The exact resistance of an element is often not important as it will not usually change over its life span except to become totally open (show infinite resistance) when defective or becomes shorted to ground (see below). In case you're curious, a large cooktop surface burner is usually in the area of 27 ohms, a small 45 ohms. A bake or broil oven element's resistance may be in the area of 20 to 40 ohms depending on its wattage.
Short to Ground
An element can also become partially shorted to ground. While this may not be enough to create a dead short and cause the element to fail outright, it can create a shock hazard. To test an element for a short to ground, an ohmmeter should be set on its highest ohm scale (1K or 10K) and tested from one of the element's terminals to the element's metal sheath. It may be necessary to rub the outer element surface with the meter probe to make a good contact. If anything other than infinite resistance is shown, replace the element.
BTW. When performing the above test with the meter set on such a high ohm scale, be sure not to be touching both meter probe's metal tips with your fingers. If that were to happen the meter may display the resistance of your skin instead of just that of the element. You can touch one of the meter leads while testing without causing an erroneous reading.
If while heating an electric range element has one spot which is glowing brighter than the mostly uniform color of the rest of the element, it is likely a sign of an impending failure. In such a case the element should be replaced to avoid possible critical failure as described above. This type of failure would likely show a short to ground if tested, which could also be a shock hazard.
If a range is giving off shocks when touched, one of its elements or its internal wiring may have a partial short to ground but not enough of a short to blow a fuse or trip a breaker. Such an occurrence should be looked into immediately as it could pose a possible electrocution hazard. Each element should be isolated from its wiring circuit and tested for a short to ground as described above.
A conventional electric range oven or cooktop element will usually either work or not (except in the case of a partial short to ground in which case they may still work normally but still be defective). If such an element is only getting warm when it should be hot, there has got to be a problem in its power supply and not in the element itself.
Plug-in elements may have an additional problem where their connecting tips insert into the stationary element receptacle on the appliance. That problem and the corresponding repair is described at the link below.
What's the difference between a stove element and a dishwasher or water heater element?
Immersion elements like used in water heaters have a higher grade of stainless steel sheath than stove elements usually do, to protect against corrosion. Such elements usually also need to have moisture proof end seals to prevent saturation of the MgO (Magnesium Oxide) insulation which would result in a short and/or possible shock or electrocution hazard.