Re: How do multimeters work?How can they read voltage?

Multimeters are devices which are used to measure various parameters of 
electric current which include, resistance, voltage and current. (Modern 
electronic multimeters can also measure other parameters such as frequency, 
capacity (of capacitors), breakdown voltage (for diodes), and check devices 
like transistors.

Since you have not specified whether you want to know about digital 
multimeters or the "traditional" analogue one's Ill restrict the discussion 
to the analog multimeters (The ones with the moving arm or wiper) since 
they are more common.

Whenever electric current passes through an electric conductor a magnetic 
field is set up around it.
(Note: Curl the fingers of your right hand with your thumb pointing 
outwards. If the direction of your thumb points in the direction of the 
flow of current. Then your fingers will point in the direction of the 
magnetic field.)
The higher the current the higher is the magnetic field setup. 

This principle is used in many electrical appliances like motors, 
generators, solenoids and of course voltmeters and ammeters.

To see this phenomenon hold a magnetic compass close to a motor or 
generator, or even an electric wire. Make sure you dont touch the 
equipment and have an adult nearby. You'll see the compass needle deflect 
from its position as you near the wire.
Another simple experiment could be done by turning a wire many times 
around a iron nail. Connect the ends of the wire to a battery. Keep the 
nail on top of a compass. This will deflect the compass needle. The higher 
the number of turns on the nail, the higher will be the deflection of the 
compass.

Getting more specific, voltmeters contain a coil having high 
resistance suspended between 2 magnets. The top of the coil is connected to 
the arm or needle of the multimeter. The ends of the coil are connected in 
parallel to the device across which voltage is to be measured. When current 
passes through the coil, it deflects away from its mean position since the 
magnetic field setup in the coil repulses the magnetic field of the 
magnets. The coil is suspended on springs (usually made of phosphor bronze 
) which control its movement. The springs deflect proportional to the force 
exerted on them (due to the changing magnetic field).
By varying the resistance of the coil, you can vary the sensitivity (and 
hence the range) of the equipment.

An ammeter is very similar to the voltmeter. The only differance is that an 
ammeter has very little resistance to the flow of electric current, and it 
is connected in series with the source of current.

Digital multimeters work on very different principles and have no coils or 
moving parts. They also usually have a higher range and better accuracy.

References:
1) The Usborne book of Science seems to have good explanations of 
electromagnetic effects which you will understand easily.
2)Fundamentals of Physics by Resnick and Halliday seems to have explained 
theories in detail, if you really require want to learn all about 
electromagnetics.
2) Progressive Physics by Bapat and Mahajan too have covered this topic.

Hope the above information is sufficient. If you need more information 
please feel free to contact me.

Nauzad Tantra