The history of ideas; the intellectual history of a group.
In literary criticism, this refers specifically to Wilhelm Dilthey's view that literature should be seen and studied as a manifestation of the spirit of its age. He argued that if the natural sciences explain events as the results of causal laws, 'cultural science' should explain events in terms of the meanings and intentions that people give them. These meanings and intentions, however, are formed by the historical and social events of the specific time and place that a work was written. (That is to say, you have to look at a work in relation to the weltanschauung and zeitgeist of the time and place it was written.)
This can also be applied to historical works of visual art, or nearly any historical (or current) construct of the human mind (government, religion, etc.).
Thanks to Siobhan for providing me with the SAMPA phonetic transcription of this word; /'gaist@sge'SiCt@/