'Zeitgeist' is a German word which is usually translated as 'spirit of the age' or 'spirit of the times'. In German 'Zeit' means time and 'Geist' means spirit. In German the word is always spelled with a capital letter as all nouns are capitalised in German, and the 'Z' is pronounded 'ts'.
The idea of zeitgeist is that there are key ways of thinking or key cultural, intellectual or political movements that influence the way a society develops. It is often cited as an opposition to what is called the 'great man theory of history' - the idea that it is certain prominent individuals, rather than the mood or tendency of society as a whole, that make history.
It is not sure who first came up with the word. The early 19th century philosopher Goerg Hegel used a similar phrase, and it may have derived from that.
It is also the name of a popular current movement - you can see details here.