They are usually a telescope on a mount that can move horizontally (left and right) and vertically (up and down). The mount has a scale where the readings are taken, usually in degrees for older models and arc minutes or arc seconds for newer models. The telescope normally has a crosshair to make it easy to align it to the points to be surveyed. The theodolite was the first tool that could do vertical and horizontal measurements in the same device.
Most modern theodolites are electronic and the readings are read from a digital display.
A transit is a kind of theodolite that can do a whole circle vertically and horizontally.