The ballistic pendulum was invented in the 18th century (Benjamin Robins, published in his New Principles of Gunnery in 1742). It uses conservation of momentum to measure the bullet speed. Shoot the pendulum with the bullet, and the momentum of the bullet is transferred to the pendulum. You measure the height that the pendulum swings up to, which tells you the kinetic energy of the pendulum after being shot, which tells you the momentum of the pendulum after being shot, which tells you the momentum of the bullet, which (if you know the mass of the bullet) tells you the speed of the bullet.

A ballistic pendulum isn't very good for measuring the muzzle velocity, since if you put the gun close to the pendulum, the gas escaping from the muzzle will push on the pendulum and give an overestimate of the muzzle velocity. For a better (but still not very good) measurement of muzzle velocity, you can attach the gun itself to the pendulum, and the swing of the pendulum will tell you the recoil momentum of the gun (which will be the momentum of the bullet and the gas escaping from the muzzle). In this configuration, it's usually called a gun pendulum, but sometimes it will still be called a ballistic pendulum.

Mechanical chronographs, measuring the time taken for the bullet to go from point A to point B, were already being used in the 18th century (by the 1760s, if not earlier). The time measurement was typically provided by the rotation of a paper or cardboard screen - the difference in position of bullet holes in a first and a second screen, together with the known speed of the screen (from the known rotation rate), would give the time. In the 18th century, the ballistic pendulum was usually more accurate than chronographs. However, chronographs were not affected by the muzzle gases, and the same chronograph could be used for a wide range of bullet masses (while a ballistic pendulum works best for a particular range of momentum values, and very large and massive pendulums are needed to measure the speed of heavy cannonballs).

In the 19th century, electromechanical chronographs were developed, and there were more accurate than their 18th century ancestors. At this point, chronographs could be more accurate than the ballistic pendulum, as well as smaller, lighter, and able to measure a wider variety of projectiles, and became the preferred method over the 19th century. Modern chronographs usually use optical detection of the passage of the bullet, and electronic timing. The modern electronic chronograph is the usual preferred method (but the ballistic pendulum was used well into the 20th century).

A third method is to use radar, measuring the Doppler shift of the radar reflection. This method was developed in the 20th century. It has the advantage that the radar can be placed behind the gun, or in front of the gun at various distances.

See C. L. Bragg et al., Never for Want of Powder: The Confederate Powder Works in Augusta, Georgia, University of South Carolina Press (2007) for a description of the ballistic pendulum, gun pendulum, and chronograph in action.