In a switching converter, the controller represents the integrated circuit which hosts all the logic gates, clock and driver to operate/control the power supply. This silicon chip features a feedback input which receives the signal from the compensator to drive the control variable (duty ratio, frequency etc.).
The compensator is the active (or passive) circuit which lets you place poles, zeroes, gains (or attenuation) to shape the open-loop response: choose the compensation strategy to force a crossover frequency and various margins (phase and gain) for specific transient response or rejection capabilities.
In analog electronics, a compensator is an active filter built around an op-amp, a transconductance amplifier (OTA), a shunt regulator (TL431) etc. In a digital world, the compensator can be coded or hard-wired to form a PID structure (for example) effectively placing poles and zeroes adequately.