This page contains a brief introduction to Vampire: the Masquerade, for those who haven't played it before.
Vampire is set in White Wolf's World of Darkness. This is very like the real world except that it is a little bit nastier, a little bit more dangerous and sleazy. More guns, more pollution, and the happy flower-seller on the street corner is probably an axe-murderer in his spare time. Also, vampires exist (not to mention werewolves, wraiths, and quite a few other weird things).
A vampire can create another of its kind by draining the blood from a human and then replacing it with its own. If a human drinks vampire blood without losing their own first, they become what is known as a ghoul; these have some of the powers of a vampire while not becoming vulnerable to sunlight, and are often used as servitors by Kindred.
Cities are usually ruled by a Prince. The Prince of a city supposedly supports the Camarilla and recognises their authority, though they often rule more or less as they wish (...as long as there isn't a Justicar in town). There is also normally a group of Primogen, consisting of the most important member of each vampire clan.
Vampires belong to one of several clans, depending upon their bloodline. The Camarilla consists of seven of these clans: the Brujah, the Gangrel, the Malkavians, the Nosferatu, the Toreador, the Tremere, and the Ventrue. Each clan has certain powers associated with it; these range from supernatural speed or strength to the ability to shift shape or the ability to vanish from sight. Other clans exist but never joined the Camarilla, and are very rare in Camarilla-controlled areas. There are also the Caitiff, those whose blood is too weak for them to claim to be part of a clan.
The power of a vampire depends upon both their age and their generation. Generation has little to do with age. According to legend, the first vampire was Caine, as in the guy in the bible who killed his brother. Those Kindred he sired were of the second generation; those created by these were of the third generation, and so on. The blood is more diluted in each generation. Most princes tend to be around the sixth generation; most newly-created vampires tend to be around the thirteenth generation.
There's probably lots of things I've forgotten to include, but I think that covers the most important things. The glossary page should explain most of the terms idiosyncratic to Vampire. If anything seems confusing, ask a GM.