Choizer (Heb. חוזר) lit. "one who reviews," pl. choizrim (Heb. חוזרים) is a title that refers to a rabbi who memorizes the teachings of one of the Rebbes of the Chabad-Lubavitch Hasidic movement. This term is not unique to that movement.
Jewish law forbids writing on the Shabbat or Yom Tov. Thus, on such occasions, when the Chabad Rebbes would deliver their discourses, certain disciples with a photographic memory were appointed to memorize every word they heard. This is known as the role of the choizer. The choizer would review the entire discourse after the Rebbe had finished delivering it with those interested to hear. This delivery was known as chazoro, a reviewing.
After consultation with other choizrim, the choizer would typically (but not always) transcribe the words that he remembered once it became permitted to do so, making him a meiniach. This transcript, known as a hanocho (pl. hanochos), was then released for public study. The unedited hanochos of Rabbi Menachem Mendel Schneerson's public addresses are available in the Sichos Kodesh, Hisva'aduyos, and Sefer Ha'Sichos series.