The letter z is conventionally used to represent a complex number, of the form z = x + iy, where i is the square root of -1, making iy an imaginary number. It is also a shorter name for the left-wing alternative news source Z magazine.
In phonetics terms, the z sound in English is a voiced alveolar fricative: Unlike the s sound, which it otherwise resembles, we produce sound with our vocal cords when we say z; this give the sibilant a buzzy quality. The sound is produced between the tongue and the alveolar ridge.
The exact sound of a z varies from country to country, but like any letter most of the sounds it can stand for are at least somewhat related. In German, Maltese and some transliterations of Hebrew it's a sort of ts sound; it's the same in Italian, except when it's more of a dz. In modern Pinyin Chinese, z is used to represent the sound that earlier schemes rendered as ts or tz, so that Lao Tzu becomes Lao Zi or Laozi. This is an unaspirated alveolar sound, Excalibre tells me; like the other languages mentioned so far, it's an affricate, which is to say it's a fricative that starts with a plosive sound. In Japanese, it is dz in zu, but otherwise a lot like the English z. In Castilian Spanish it is pronounced as a th, but in the Spanish of Andalucia and South America (and also Basque) it is pronounced more like an s. In Scottish names like Dalziel and Menzies, the z takes the place of the extinct yogh, which stood for a y or a gh sound.
Ž in various languages (that's z with a hacek), zh, Chechnya's z with a line through it and the Russian Ж are all pronounced as a voiced palato-alveolar fricative, much like the s sound in the English pleasure and leisure and the French j. Other accented zs include ż and ź, both used in Polish.
With thanks to Gritchka for much of the information on phonetics.