A transducer is defined as 'an electronic device that converts one form of energy to another'. Examples of transducers are microphones, thermometers, antennas, etc. This is what the pickup does for your violin: its transducer converts the sound of the strings into electricity which is further processed by the amp.

Most pickups are made from piezoelectric material. Other types of transducers use magnetic coils to convert the sound.

Piezoelectric

A piezo is a crystal that converts pressure to energy. There are a few advantages to using piezo transducers: they are small so they can be mounted in the bridge, they are very sensitive, so they will pick up every nuance of the sound. Piezo transducers offer a great sound when amplifying the lower strings on the electric violin but can give unsatisfying results on the upper strings, especially if you're not using a pre-amp. They also pick up the body resonance that can create feedback, so you can't use them at the highest volume.

Electromagnetic

Magnetic coil pickups are the same transducers as those used on electric guitars. They are the exact opposite of the piezo transducers: you can use them at high volumes, but they will never give you the detail and complexity of the piezos. The technology behind this kind of transducers is simple: a metal core is wound with wire, thus creating a magnetic field that can be broken by the vibrations on a cord. When this happens a certain voltage is induced in the coil. An important consequence is the fact that you can only use metal strings with magnetic coil pickups.

You can find detailed information and links about most Electric violin transducer manufacturers on dedicated pages.