Zembekiko is both a type of dance and a type of music in a manner analogous to how "cha-cha" can describe the rhythm of a song and the dance itself. It belongs to the Rembetiko group of Greek dances, only where the more popular Greek dances involve people linking hands and dancing set steps in a circle, this is a solo and improvisational dance performed traditionally by men. Zembekiko music is generally set at a measured, steady pace (about 60 beats per minute, usually a touch slower but rarely much faster) and features the bouzouki, though percussion and bass are usually included.
In addition, the songs are almost invariably about troubles in love and heartbreak, and so the dancer of the zembekiko acts out the character of someone who has had a bit too much to drink in attempting to drown his sorrows and gets up to express his melancholy in dance. In token of this, watchers may put a glass of wine or a shot of ouzo on the dance floor, and if they do, the dancer usually acts out a love-hate relationship with the liquor, approaching avidly and drawing back, both craving more and knowing he should not have it. A good zembekiko requires grace, balance, creativity and rhythm, and a truly impressive one requires agility and flexibility as well, but the main point of the dance is, as always, for the dancer to enjoy himself.
As for spectators, these days in very crowded tavernas, they may join in with their own zembekiko, and any two given dancers may dance around each other and integrate their steps or ignore one another in favour of their own dances; traditionally, however, one allows the dancer his solo time in the spotlight. Other spectators surround the dancers either in a squatting or kneeling position and clap out the rhythm of the song while giving shouts of encouragement ("Opa!"). Often, watchers throw flowers, petals or money at the dancers' feet -- or even break plates or other things (all of which I have personally witnessed at various tavernas).
The origin of the zembekiko is unclear, but some Internet sources state that it is a war dance of the Zeibekides, a tribe that originated in Thrace, and others say that the music may have originated in the chants of the Byzantine church.