Good question! Historically, there are two mechanisms that keep the pendulum swinging: weights and springs. If you've ever seen a grandfather clock, you've probably seen the weight mechanism: hanging from the main mechanism of the clock are weights on chains; as the pulley swings, the weights slowly lower and transfer their potential energy to the pendulum, keeping it swinging. Once the chains are fully extended, someone has to manually rewind them. It's also possible to use coiled-up springs to power the pendulum; like the weights, once the springs are unwound, they need to be recoiled manually.

Nowadays, though, there are some pendulum clocks that do use electricity to keep the pendulum swinging (though that's not as old-fashioned as weights or springs!)

Let me start by saying that mechanical clocks cost very, very little energy since it is doing practically no work. That it tells time for us is convenient, but doesn't cost energy in principle. An ideal mechanical clock (no friction) would cost no energy to run, only energy to start. Newton's first law of inertia tells you that if you pull a pendulum and let it go, it will continue to go back and forth until friction forces rob it of its energy (initially supplied by you to lift the pendulum arm in the first place). That said, here's a brief description of how mechanical clocks work:

Mechanical clocks work by having a weight connected to a string, then wrapped around the shaft of the first gear. The weight constantly pulls downwards on the gear, trying to turn it. There is generally an additional piece of wood that sticks in one of the gear teeth preventing it from turning (holding the weight back) which is connected to a pendulum. The pendulum's length is set so that it swings from one side to the other in 1 second and each time it gets to the opposite side it lets the gear tooth turn once.

The energy to swing a pendulum in a mechanical clock can be any form, though commonly mechanical clocks will use a spring or a weight. If a spring is compressed, it will do work, like move the pendulum, as it decompresses. Likewise, a weight will want to fall towards the ground, and this can also do work.

The source of energy can be anything, but the crucial piece of machinery that converts this energy into time units is the escapement. Also, the escapement is what causes the ticking noise! For example, a grandfather clock may have weights, and the escapement converts this energy into ticking.

So why pendulums? The reason that pendulums are used is because they make clocks more accurate (if it's not moving and not tilted sideways). When a pendulum swings, The time it takes for one full swing only depends on the length of the pendulum. Interestingly, it doesn't matter how heavy it is or how far it's swinging (start from 5, 10, 20 degrees), it will always take the same amount of time. Because of this, when you build a pendulum of a certain length, you can calculate how many seconds it takes for one swing, and from that you can count time! (Or if you want ticks every second, you know how long you need to build your pendulum.

Old-fashioned pendulum clocks have either a stretched spring or a pulley with a weight to give them the energy to run.

Energy equals force times distance, and this comes from a stretched spring or a pulley with a weight.

People had to wind up the clocks every day or week, to give energy to stretching the spring or moving the pulley and weight back.

The pendulum measures the length of time necessary to make one complete swing, which is usually one second in a standard clock. The pendulum swings to its high point, then is pulled back by gravity to swing to the opposite point, and vice-versa. Of course, energy is lost with in the form of sound and heat due to friction with each swing - I believe that this is stored in a mechanical clock in the forms of springs of some sort, but I am not enough of an engineer to tell you exactly how. Most of the pendulum-driven clocks that I know about actually are electric, and use electromagnets to recharge the pendulum's swing so that it does not lose any net energy, and use the pendulum as a measure of time.