R4 is the emitter stabilising resistor.

When collector current flows it causes the transistor to heat up.
This causes the base current to increase which in turn causes the collector current to rise.
This rise in collector current causes the temperature to increase even more, and the base and collector currents continue increasing.
This behaviour is called THERMAL RUNAWAY and will destroy the transistor.

If we consider the emitter/base junction as a diode as shown in the right hand diagram we can see that the base (anode) voltage is fixed by R1 and R2.

If the collector current tries to rise due to heating, then the voltage across R4 will try to rise, making the emitter (cathode) more positive.
This would reduce the voltage across the junction (diode) making it less forward biased and reducing the base current and hence the collector current, which was trying to rise.

Therefore the circuit has been stabilised against thermal runaway.

However, if an ac signal is applied to the base, the varying collector current will cause a varying voltage across the emitter resistor.
This voltage will follow the base voltage.This means that the base/emitter (anode/cathode) voltage will be constant instead of the base varying with respect to the emitter.
To fix this, the emitter is joined to earth, as far as the ac signal is concerned, by the emitter decoupling capacitor.