The term 'signal fault' very rarely means a fault with a signal itself, but generally with a part of the system which interlocks with it.
Railway signalling is designed to fail safe, that is if there is a fault, then the system will default to a state that will stop trains and prevent accidents.
There are six systems involved. Two you're aware of, the signals and points. The third is the control system used in the signalbox itself, and then there are track circuits and axle counters that give the signalman the location of every train, Advanced Warning System (AWS) which warns drivers of the indication of the next signal, and Train Protection & Warning System (TPWS) that will automatically activate train brakes in certain instances.
All these systems work together and a fault with one can affect the other.
Signals are positioned along the line to keep trains a safe distance apart, and depending on how busy the stretch of line is, are used to regulate the traffic. The different coloured lamps tell the driver how far ahead the track is clear for. If one of those lamps fails (and they have multiple filaments to reduce this), the next most restrictive light will show, thus affecting the information the driver sees.
Points are used to move a train from one line to another. They are operated by the signaller (often many miles away from the location) and generally use an electric motor to move the points across. Apart from the drive system, there are a number of 'proving' circuits involved, which indicate to the signaller that the points are fully across and safely locked into position. If there is an obstruction (e.g. tree branch, stones, litter), these circuits will fail (as they cannot detect that the points have moved), and this will set the signal that protects the points to red, thus stopping an approaching train.
Track circuits and axle counters tell the signalman where a train is. Each circuit or counter is isolated from its neighbours. As a train enters each section, it short-circuits a relay, this will then set the signal behind the train to red. This is why if you are near the back of a train, it will seem that you always pass red signals, but you can't see the red signal at the back which protects your train! If this circuit gets a false current (metal objects across the track are a favourite to cause this), then the relay will short-circuit and set the signal. If a track circuit shows occupied when it shouldn't, that will also lock the points and signals that feed into it.
The control panel in the signalbox consists of all the controls that the signaller needs to operate signals,points and level crossings. These generally cause few problems, but occasionally a fuse or bulb will blow giving a false indication which needs to be investigated.
AWS is a system which uses permanent and electro-magnets and is used to inform the driver the state of the upcoming signal. There is either a bell or buzzer in the cab of the train, the bell sounds if the signal is 'clear', and a buzzer if it is 'restrictive'. The driver can ignore the bell, but MUST cancel the buzzer by pressing and releasing a button within six seconds. If he doesn't, the train brakes will come on. If there is a problem (power issues affect AWS most), the permanent magnet controls the 'restrictive' part, and this will make the driver run slower as they cannot be sure that the line ahead is as clear as they would expect.
TPWS is similar, this will activate if it detects that a train is running too fast and also if approaching a signal set at red and will automatically operate the brakes in these instances.
All these systems interlock together, and are controlled from a signalbox, often very remote from location. This means that are literally thousands of miles of cables linking them all together, which can be affected by the weather, rodents, vandalism (there was an instance this weekend where vandals smashed open the back of a signal, smashed all the bulbs and ripped out all the wiring) and of course, any component, no matter how well maintained, can fail or even be duff when new. Added to those miles of cables are millions of fuses, relays, capacitors, gears, cogs, motors, levers and other items and although they are all inspected and tested at frequent intervals, nothing is perfect.
I hope that at least gives some idea as to what can cause these problems and why services are stopped when they do.