The bouzouki is a string instrument of Anatolian origin. It's a type of lute and as such is one of a large family instruments used in every place between northern India and southern Italy. Its name comes from the Turkish bozuk (atesh tells me that means "broken" but neither of us can explain the connection).
The instrument in its traditional form has three double metal strings, E B E, and that's the form most widespread in contemporary Greek music. The four-string variant, D G B E, is a modern innovation. Its shape resembles a mandolin but the bouzouki is significantly larger and plays an octave lower. Its metal strings and the playing style mean a pick is required.
Traditionally the bouzouki is associated with Greek music. However, its origins are farther east--the bozuk was a traditional Turkish instrument but we don't know whether the Turks got it from the local ethnic Greeks first or whether they developed it--and it was introduced to Greece in more or less its current form in the 1920s. Its association with the lowest strata of Greek society and its use in the rembetiko, the musical genre commonly associated with drug addicts and criminals of the time led to it being sometimes tolerated, sometimes illegal but never respectable. This changed in 1949, when the composer Manos Hadjidakis, known as darling of the elite, first shocked said elite by presenting the bouzouki as a valid orchestral instrument and then started using it regularly in his compositions. Following the adoption of its hallmark instrument by the mainstream the rembetiko went into decline but the bouzouki continued as a mainstay of popular music from then until the present.
The bouzouki has, strangely enough, also become known as a Celtic instrument. The four-string version is not often used in Greek music but contemporary Celtic musicians use only the mandolin-like four-string version. The date at which this strange exchange between centuries-old musical traditions occured is not certain but it probably occurred in the mid to late 1960s and no later than the early 1970s. The instrument known as an Irish bouzouki is not a pure bouzouki, not even a four-string one, but is rather a cross between the Greek bouzouki and the cittern.
You may encounter "bouzouki music" tapes or discs on sale in tourist shops in Greece and stores abroad. While these will give you a reasonable idea of the instrument's sound, the songs on most of them are easy listening and don't present it at its best. One of the most widespread songs on such compilations is the famous theme from Zorba the Greek where the bouzouki is the lead instrument (in fact, this 1964 composition defined the outside world's idea of "bouzouki music"). This is a poor example of the kind of music the bouzouki belongs in but most of them will have a virtuoso playing the instrument so you can get a good idea of its capabilities. Another "artsy" arrangement but more traditional composition you might often find is the theme from Evdokia which combines the bouzouki with its little brother, the baglama. If you happen to notice the names Loizos, Nikolopoulos or Zambetas in the credits, you can be assured of quality playing and okay music.
A cheap bouzouki can be bought for as little as about USD 100-150 in Greece but it won't be much of an instrument. You'll pay USD 400 and upwards for a new quality instrument though, as with any musical instrument, there are bargains on the used market. And don't let anyone tell you their bouzouki is not Greek; if it's not Greek, it's not really a bouzouki.