If it was launched at the Kennedy Space Center in Florida, there are ten of these Contraves-Goerz Kineto Tracking Mounts spread around the complex. Contraves-Goerz is now a part of the L-3 Brashear company. Multiple cameras are attached and all the cameras on each mount are precisely aimed on parallel optical axes. Other launch facilities use similar, if not identical, hardware, though perhaps in fewer total numbers per facility.

The earliest similar types of mounts used to track and film rocket launches were based on early naval antiaircraft gun mounts. Some of these early versions had spotting telescopes used to optically aim the contraption.

Today a lot of the tracking functions are automated or semi-automated, but back when it all first started the skill of the operator was responsible for keeping the camera array pointed in the right direction. Just as with those old anti-aircraft guns, the thing was aimed by using sets of control wheels, one set for altitude and one set for azimuth. These control wheels are similar to what one would see on large, crane mounted cinema cameras on a movie set during the same time period. The rocket tracking control wheels had both higher and lower ratio gearing that made both coarse and precise adjustments possible. A full 360° turn of the precision wheel might only move the mount a fraction of 1°!

Similar (near identical) systems are used to track and record other high speed flying objects. This would include military hardware during testing and developement (planes, missiles, even large artillery projectiles such as those launched by a tank or larger artillery guns, etc.)

You can read all about the (non-classified parts of the) workings and performance of these systems here.

This document from NASA explains how they are used at the Kennedy Space Center.