The local oscillator is voltage controlled (it is a VCO).
This means that its frequency is controlled by varying a DC voltage input.

The output signal of the VCO is fed back to a phase detector via a buffer.
The buffer isolates the VCO from the loading caused by the detector and external circuits.
It avoids "pulling" of the oscillator frequency.

If there is no reference input signal then the VCO will oscillate at its natural "free running" frequency.

The other input to the phase detector is the reference signal, which we wish to lock the frequency of the VCO to. 
If there is a difference in frequency or phase between the two inputs then an error signal is produced at the output of the phase detector.

This error signal is fed to the VCO via a filter and a DC amplifier to produce an error correcting voltage. 
The filter is a low pass type which determines the range of frequencies over which the VCO can vary.
The DC amplifier amplifies the DC voltage level to a value suitable to control the VCO.

The error correcting voltage may be either negative or positive depending on whether the VCO frequency is higher or lower than the reference frequency.

The effect of the error correcting voltage is to pull the VCO back to the same frequency as the reference frequency, and in phase with it.

If the input signal is an FM signal, then the VCO follows the deviations in frequency caused by the modulation, and the DC output of the DC amplifier is the demodulated audio signal.

The PLL can also be used to keep an AC motor at a constant speed.