The primary divisions among Jews beginning in the medieval period were Sephardim and Ashkenazim. Its quite clear however that dividing Jews into only these two categories is incorrect and oversimplification. The term Sephardim is derived from the Hebrew word for Spain, Sepharad.
The term is first used in the Bible, but its connection to Spain or more correctly the Iberian peninsula at this time is unlikely.
Eventually however the term was used for Spain, and hence Sephardim reflects Jews who were or descend from Jewish communities living in Spain and Portugal.
The existence of Jewish communities in Spain dates to the early centuries of the Common Era during the Roman period. Tradition among Spanish Jews dates the existence of Jewish communities from the time of the destruction of the Second Temple in Jerusalem in the year 70 CE, while some date the existence of communities back the period of the Babylonian exile.
The Jewish communities of Spain gained prominence in the Jewish world during the rise of Moorish rule in Spain, a subject which deserves its own series of posts. The historic Jewish communities of Spain and Portugal were eventually expelled in the 15th century which led to a Sephardic Diaspora Spanish and Portuguese Jewish communities in the Ottoman empire as well as in various places in Europe and eventually to the New World.
Today four communities are correctly categorized under the Sephardic label, though the term Sephardic is often misused to designate other communities that were from North Africa, the Arabian peninsula, or present day Iraq and Iran.
The four communities are Spanish-Portuguese, Judeo-Spanish, Moroccan Spanish, and Syrian communities. The number of Jews under all these categories has drastically fallen though communities of each continue to exist.
The Ashkenazim are Jews which originated in communities in Ashkenaz, the Hebrew name for Germany. The term Ashkenaz appears in the book of Genesis in an early genealogy detailing the descendants of Noah, and it likely designated a region in Europe in later Jewish tradition, though at what time it was specifically connected with Germany is unclear.
Like the term Sephardim, the term Ashkenazim has been extended to Jewish communities in other regions, though perhaps less inappropriately than the former term. The term correctly covers those Jewish communities that lived in medieval France and Germany. It was later used in referring to Jews in Poland, Russian, as well as other communities of Eastern Europe.
The Ashkenazim today typify the communities that most people associate with Jews. In the United States, the overwhelming majority of Jews are Ashkenazic background, though interestingly, the first four Jewish communities in the United States were founded by Spanish and Portuguese Jews.