What is the difference between a galvanometer, an ammeter and a voltmeter? I'm looking for basic differences b/w the three with respect to nature of the circuit as well i.e either it is connected in parallel or series, and also the reason as to why they're connected the way they are? In what cases are these three used?
 A: Galvanometer is a basic electromechanical instrument for measuring small currents. Old style analog ammeters and voltmeters are built around galvanometers. 
It is an electromechanical instrument because it translates electrical current into a position of a mechanical pointer. 
The heart of a typical galvanometer is a coil, which turns in the magnetic field of a permanent magnet, when a current flows through it. To measure the current in a circuit, the galvanometer coil has to be inserted in the circuit. 
Galvanometers are typically made very sensitive, i.e., the deflection of a pointer per unit current is relatively high and, therefore, a full scale current is relatively low, commonly on the order of $100\mu A$, which limits the direct measurements to relatively small currents.
Ideally, the resistance of the galvanometer coil should be zero, in which case its insertion would not change the current in the circuit. In reality, the resistance is not zero, so, for accurate current measurements, the resistance of a galvanometer has to be small relative to the resistance of the rest of the circuit.
To overcome these limitations, a relatively small resistor (shunt) is connected in parallel with the coil, so that only a small fraction of the current in a circuit flows through the coil. Since the ratio of the coil resistance and shut resistance is known, the actual current in the circuit (a sum of the shunt current and the coil current) could be calculated. So, with a number of switchable shunt resistors, a galvanometer could be converted to an ammeter. 
A galvanometer can also be used to measure voltage. If the coil of a galvanometer is connected between two points of a circuit, say, A and B, with some voltage between them, a fraction of the current in the circuit would branch off and flow through the coil. Since we know the resistance of the coil and can measure the current flowing through it, we can calculate the voltage between A and B.
That, however, would work well, only if the resistance of the circuit between A and B was much smaller than the resistance of the coil, so that only a small fraction of the current would branch off to the coil. Otherwise, the current in the circuit between A and B would decrease too much and the error would be significant. Also, if the voltage between A and B is high, the current flowing though the coil could exceed its limit.
To overcome these limitations, a relatively large resistor could be added in series with the coil. In this case, only a small fraction of the circuit current would be diverted to the coil. Knowing the ratio between the resistance of the series resistor and the resistance of the coil and measuring the current, we can calculate the voltage between A and B (a sum of voltages on the series resistor and the coil). So, with a number of switchable series resistors, a galvanometer could be converted to a voltmeter.
In summary, both ammeters and voltmeters, based on galvanometers, measure the current flowing through a coil of a galvanometer. To measure wide range of currents, the resistance of a galvanometer is reduced by adding shunt resistors in parallel with the coil. To measure wide range of voltages, the resistance of a galvanometer is increased by adding resistors in series with the coil.
A: Well, V.F. has provided a comprehensive look at how a galvanometer work and how one builds ammeters and voltmeters on that basis, but I'd like to offer a high-altitude approach as well.


*

*An ammeter is any instrument for measuring current.

*A voltmeter is any instrument for measuring electric potential difference or EMF.

*A galvanometer is a particular type of analog ammeter using induction in a coil to deflect a needle against the resistance of a spring. They were often used as one component in building wider range ammeters and voltmeters.


If you go down to the hardware store, automotive store, or Radio Shack (if you can still find one) you may find a variety of cheap multimeters on sale. If you find one that has a mechanical needle it is probably implemented on a galvanometer, but most of them will have some kind of digital display and these are build on solid-state electronics; their working is quite different from the ones that V.F. discusses.
