The theodolite is a precise instrument used to measure horizontal and vertical angles. It has greater accuracy than a magnetic compass, able to measure angles to within 10-20 seconds. The main components are a horizontal circle to measure horizontal angles, a telescope that can rotate vertically and horizontally, and spirit levels. Measurements involve setting the instrument over points and using the horizontal and vertical circles to measure angles to other points using techniques like repetition or reiteration. The theodolite is used for tasks like traversing, measuring deflection angles, and computing latitude and departure distances.