The bouzouki
The main instrument of Pireas-style Rebetiko, without a doubt, is the bouzouki. Originally developed from the tamboura and saz and various other lute-like instruments, the first bouzoukis had six strings. Since these are in three pairs, this type of bouzouki is called a trichordo or trixordo. (That's Greek for three strings.)
Very early trichordos had movable frets, tied at the back of the neck, so that the player could adjust the notes that were playable to fit the makam or scale of the piece being played. However, as Rebetiko developed, it moved away from the classical Ottoman makam based music, and settled on its own versions of the scales. These often have the same name as the Ottoman makam, but are subtly different. The bouzouki then became a fixed fret instrument.
There is some debate as to whose idea it was to add a fourth course of strings to the bouzouki, and change its tuning to match the top four strings of the guitar. There is no doubt at all that the player who popularised this version of the instrument was Manolis Chiotis. Not everyone regards the change as an improvement...
Chiotis was also a very early player of amplified bouzoukis, which were necessary given the size of audience he was getting. The tetrachordo bouzouki, fed through reverberation units, effects units and into powerful amplifiers, is more or less the de-facto standard nowadays, along with the Irish bouzouki (also usually amplified), and a quick Google search will get you as much information about it as you wish, possibly more. Recent developments have included a MIDI bouzouki which could be used to control whole racks of synthesizer equipment. The web site for this has now vanished, so perhaps it wasn't selling.
My personal preference is for acoustic trichordo, so if you want to find more information about tetrachordo bouzoukis, try here.
Acoustic trichordo bouzouki
My own bouzouki looks very plain. This is because the luthier who made it, Karolos Tsakirian, built it as a re-creation of a bouzouki his grandfather built for Markos Vamvakaris. The bowl is a little smaller than that of a modern bouzouki, and was made on the same mould as Vamvakaris's bouzouki. The scratch plate inlays on the sound board are made of rosewood. It sounds like this...
Of course, most bouzoukis are decorated a lot more than mine. You can see some wonderful workmanship in the Skeezo Gallery. The Remoundas bouzouki with abalone peacocks is particularly impressive. Of course, ornamentation does not make a bouzouki sound any better, and the sound is the important thing.
Tuning
The strings of a trichordo bouzouki are tuned like this...
Note that this diagram shows the notes an octave higher than the actual notes, to avoid having a lot of messy leger lines.
Here is a recording to assist anyone who needs to tune a trichordo bouzouki.