It helps to have an elementary understanding of what goes on inside a clock in order to appreciate why a clock might stop for no apparent reason or why it won't run for a full 8 days anymore. So I'd urge you to read the page on 'how pendulum clocks work' before reading what follows. That introduction will also help you identify the names given to different parts. Once you appreciate what a delicate instrument a clock is and how reliant it is on there being sufficient power reaching the escape wheel you're well on the way to anticipating why a clock might stop working.
First, if there is any drag or other resistance to impede the rotation of the wheels, then the power reaching the escape wheel will be diminished. And when the power falls below a critical point, the teeth of the escape wheel will cease to provide the essential impulse to keep the pendulum swinging. Typically, resistance comes in the form of dirt accumulating in the pivot holes or the gradual thickening over time of the thin oil used to lubricate them. Pinions might also become dirty or rusty and brass teeth can corrode with the formation of verdigris.
Two other things commonly happen to clocks if they are never serviced.
First, when a pivot turns in a dry pivot hole, it causes wear. Curiously, it's not usually the hole that wears first but the pivot. This is because debris and dirt collecting in the hole is forced into the soft brass and remains there grinding away at the steel pivot as it rotates, leaving it misshapen. Of course the steel pivot, being harder than brass, also affects the brass. So, second, the round pivot holes can become oval in shape, particularly the bigger ones near the mainspring.
This happens because the forces of the mainspring act in one direction while the next wheel in the train is reluctant to turn; so the two competing forces drive the pivots apart, forming an oval in the pivot hole. This is dangerous if ignored because as the wheels gradually drift apart, so only the tips of the teeth engage with each other and then one day, usually while you're winding the clock and increasing the power in the train, the teeth bend and slip uncontrollably. Replacing stripped teeth is an expensive repair and a complex repair is not desirable anyway as the integrity is impaired. Anyway, it might be cheaper to source another movement.
Another common problem arises when a clock is moved from one position to another. If you have to carry a clock any distance at all, first carefully remove the pendulum. If you don't there is a risk that as it bounces back and forth erratically while in transit, it will upset the position of the crutch it hangs from which must be carefully set in relation to the escapement in order to put the clock 'in beat'. You'll find more about setting a clock in beat elsewhere on this site. It's not a difficult job to put right but it takes some experience and is an irritation for the owner because the clock not only sounds sick, but it may well cause your clock to stop working after just a couple of minutes. What's happening is this: the pendulum falls one way and gets a good impulse from the escape wheel as it reaches its point of return, but when it falls the other way, the pallet leaves the tooth of the escape wheel before it reaches its naturally peak and so gets no such impulse to drive it. So the pendulum is now working on half power and will inevitably stop if there is any other impediment.
Such an impediment could simply be a slightly short or mis-shaped tooth on the escape wheel which does not give a sufficient impulse the other way. Normally, one damaged tooth will not stop a clock but if it's running dry or in a mixture of dirt and congealed oil, with rusty pinions and corroded teeth that barely engage because of mis-shaped pivots turning in worn out pivot holes, and all the time out of beat ...?
Well, it's a wonder it ever worked at all.