This question was asked w.r.t the Electricity topic where there was an experiment that demonstrate current is due to the flow of negative charges( electrons)
In terms of the how they work, there is no difference. Infact an ammeter is a calibrated galvanometer use to measure current in Ampere values. However a galvanometer does not measure in ampere values. That’s where the difference lies.
A definition of galvanometer is given below.
GALVANOMETER [galvanometer] , instrument used to determine the presence, direction, and strength of an electric current in a conductor.
( http://www.encyclopedia.com/html/g/galvanom.asp )
This is the definition of Ammeter from the same source.
AMMETER [ammeter] , instrument used to measure the magnitude of an electric current of several amperes or more. An ammeter is usually combined with a voltmeter and an ohmmeter in a multipurpose instrument. Most ammeters are based on the d’Arsonval galvanometer and are of the analog type, i.e., they give current values that can vary over a continuous range as indicated by a scale and pointer or digital readout.
Another good source to visit is Answers.com, there the answers are complied from various sources into 1 page. Very convenient location for getting an overview of the answers in a page. I reproduced here some of the interesting details on galvanometer.
History of Galvanometer
The term “galvanometer” derives from the surname of Luigi Galvani. Many early applications of galvanometers for measuring and recording are associated with William Thomson (Lord Kelvin). The earliest galvanometer was reported by Johann (Johan) Schweigger of Nuremberg at the University of Halle on 16th September 1820. André-Marie Ampère also contributed to the development of the galvanometer.
And on the sensitivity of galvanometer
The sensitivity of modern galvanometers ranges up to 0.04 in. (1 mm) of deflection, on a scale 40 in. (1 m) distant from the mirror, for a current of 0.00001 microampere. Such a galvanometer may have a coil resistance of 800 ohms and a critical damping resistance of 100,000 ?. The voltage response of this instrument amounts to 0.04 in./microvolt (1 mm/?V) at critical damping. A galvanometer designed for voltage sensitivity, has a coil resistance of 20 ?, a critical damping resistance of 30 ?, and a response of 0.04 in. (1 mm) for 0.05 ?V in the critically damped circuit. It will be seen from these examples that a large response to current is associated with large coil resistance and high critical damping resistance, whereas voltage response is associated with low coil resistance and low critical damping resistance.
( Source : http://www.answers.com/topic/galvanometer )