essay immenent, prepare for head buster...
the multimeter will have a dial, usually split into 3 parts, one for voltage (in volts either dc or ac), one for current (in amps) and one for resistance (in ohms). you can choose different settings depending on what you are measuring, for 12v dc you should select DC voltage and '100', that means the voltage display will measure up to 100v, and on a 3 digit screen the readout will display actual voltage, if you chose 1000v instead then a 12v reading would show as 1.2 on a 3 digit screen. this can confuse you so you need to select the correct rating before you start to measure any voltages, DC is battery voltage, AC is mains voltage. the instructions that come with multimeters are usually fairly easy to understand so read them.
as for the demister, the element on the window will have a single connector on each side, it will connect at one side to recieve power, and at the otherside to get an earth, this basic principle is how you should view all curcuits on the car, the curcuit begins with the battery positive terminal and ends when the curcuit touches the chassis of the car (earth). to find faults in curcuits you simply trace the curcuit path looking for a break in the cucuit that is preventing power from the battery from reaching the earth of the chassis.
the circuit for the demister will begin at the battery with a permanent live, this then passes through the ignition so that the circuit is only powered when the ignition key is turned, the ignition live is then routed through the fuse box, then to the relay for the demister, then through the boot pins, then to the demister itself (positive connection) through the demister, then out at the negative connection, then back through the boot pins and finally to the chassis (earth). thats quite a journey for the electricity, and the possibilities for where this is going wrong can be anywhere along the curcuit. the trick to fixing the problem is finding the break.
the best way to find a break is to test the easy bits first. on a demister you should use your multimeter to determine if power is making it to the demister. turn the key in the ignition, then switch on the demister, turn the multimeter onto voltage setting 100v, then touch the positive (red) terminal of the tester on one of the connections either side of the demister. at the same time touch the black negative terminal of the multimeter onto a good earth such as a big bolt head holding one of the back seats on. if the multimeter shows 12v you have a live, if not try the other connection at the other side of the demister. if neither connection on the demister is showing 12v dc then you know you have a problem in the power supply. if you do have a voltage of 12v dc at the demister then you know you have a problem in the earth circuit.
lets assume you have no power going to the demister, you should trace back until you can find power. the best place to start is the fusebox. find the fuse that is for the demister. remove it. now touch the positive terminal of the multimeter on one of the metal connections that the fuse pushes into. at the same time touch the negative black terminal of the multimeter to the car body, door hinge pins usually work. if there is no voltage touch the pin on the other side of the fuse connection. still no power, cry, if you do have power then you know the live is getting as far as the fuse box.
assuming you have a live at fusebox you should test for power entering and leaving the relay. this requires you to consult a wiring diagram to find the pin numbers for the relay, remove the relay and test for a live coming from the fusebox. if the power isnt getting to the relay then you have a broken wire between fusebox and relay, if power is getting to the relay you should test to see if power is leaving the relay. to do this easily you can open the boot and test the demister live pin on the back on the car (not on tailgate) to see if it is live, if it isn't then you aren't geting power from relay.
assuming you havent got a live coming from relay, you can assume either the relay is faulty, or the switch is faulty. the relay will only pass live to the demister when the demister switch is used. a faulty switch, or relay, or loos wire inbetween them, would all stop the power from getting to the demister.
if power is getting to the boot pins you should try cleaning the pin on the tailgate that connects to the live, make sure the spring is working and the metal is clean, a wire brush or piece of sandpaper can work wonders.
now i'll explain the procedure assuming you had power in the first test. you need to test the earth pin for the demister with the multimeter. the earth pin is shared for rear wiper and demister, thats why there are only 3 pins on tailgate. change multimeter to resistance. when you find the earth pin you should touch negative terminal ot multimeter to it. touch positive terminal of multimeter on the live pin for demister. if the multimeter says '1' then you have a circuit break in the tailgate itself, any other value shows the circuit in the tailgate is ok.
final check is the earth pin on the back of the car (not tail gate but under it) using the resistance meter see if there is a circuit from the earth pin to the chassis. if the meter reads 1 then you have a break between the earth pin and the chassis.
well there it is hope this makes sense, it would be easier to show you but i guess you get the idea. good luck