Ashkenazi is one of two major cultural labels that can be applied to a Jew, the other being Sephardi.
Ashkenazi Jews are those whose ancestors migrated from Germany to Eastern Europe during or shortly after the Crusades. In the 17th century, many of these Jews migrated back to Western Europe, spreading Askenazi Jewry across most of Europe (with the notable exceptions of Spain and Portugal).
Although Ashkenazi and Sephardi Jews observe most of the same traditions, there are some small variations in customs, such as what foods are permitted to be eaten during Passover, and what tunes accompany portions of the liturgy.
One's designation of Ashkenazi or Sephardi comes from the designation of one's father. Ashkenazim (the plural form used to refer to a group of Ashkenazi Jews) currently make up the vast majority of Jews in the world. About 80% of Jews are Ashkenazi, although the numbers are split more evenly in Israel, due to migrations of Jews from Arab and African nations, which are Sephardi.