Beth Midrash (Hebrew: בית מדרש; also Beis Medrash, Beit Midrash, pl. batei midrash or botei medrash) is a study hall (literally a "House [of] Interpretation" or "House [of] Learning" in Hebrew). It is distinct from a synagogue, although many synagogues are also used as batei midrash or vice versa.
Generally, there are either benches or chairs, and lecterns (shtenders in Yiddish), or tables, on which books are placed, and chairs for seating.
A characteristic beth midrash has many hundreds of books, including at least several copies of the entire Talmud, Torah, siddurim (prayer books), Shulchan Aruch, Mishneh Torah, Arbaah Turim and other frequently consulted works.
In modern times, "batei midrash" are typically found as the central study halls of yeshivot or independent kollelim, both institutions of religious study. The location and institution of study are often interchanged, so in popular parlance, yeshivot are sometimes referred to as batei midrash. A beth midrash may also be housed in a synagogue, or vice versa. In antiquity, this is a matter of debate (see below). Many batei midrash originally serve the community but attract a yeshiva in the course of their existence.
Early rabbinic literature, including the Mishnah, makes mention of the beth midrash as an institution distinct from the beth din and Sandhedrin. It was meant as a place of Torah study and interpretation, as well as the development of halakhah (the practical application of the Jewish Law).
The origin of the beth midrash, or house of study can be traced to the early rabbinic period, following the destruction of the Temple in Jerusalem in 70 C.E. The earliest known rabbinical school was established by Rabbi Yochanan ben Zakkai at Yavneh. Other official schools were soon established under different rabbis. These men traced their ideological roots back to the Pharisees of the late Second Temple Period, specifically the Houses of Hillel and Shammai, two "schools" of thought.
By late antiquity, the "beth midrash" had developed along with the synagogue into a distinct though somewhat related institution. The nature of the connection between the "beth midrash" and synagogue is related the question of rabbinic authority in late ancient Judaism -- a matter of considerable debate among scholars today.
For more information, see George Foot Moore's Judaism, as well as the more recent works of Jacob Neusner. Also, L.I. Levine's The Rabbinic Class of Roman Palestine, as well as the relevant articles in D. Urman and P. Flesher's edited volume, Ancient Synagogues: Historical Analysis and Archaeological Discovery.