According to the New York Times, a conductor's stick is called a "baton." Batons are primarily used with bands and orchestras. Directors of choruses are less likely to work with them.
A baton is typically held in the right hand and controls how fast the musicians play. The beginning of each measure is indicated with a downward movement, known as the "downbeat." In addition to keeping time, a baton is used to specify how music should be performed, such as smoothly, disconnectedly, subtly or harshly.
Many conductors use the left hand to show details. For example, sometimes it indicates when instruments are to start playing. Other times it helps manage the volume of the music. Conductors tend to develop individual techniques for left hand signals.