How are side bands produced?

Hint: To give the proper solution of the given question, we need to primarily get briefed up regarding Amplitude modulation.
Amplitude modulation $\left( AM \right)$ is a modulation technique that is generally used in field of electronic communication, most commonly for transmitting messages with a radio carrier wave and in it, the amplitude of the carrier wave is varied in proportion to that of the message signal, such as an audio signal.

Complete step by step answer:
Side bands could be generally said as the product of amplitude modulation. A modulating signal of frequency $\left( {{\omega }_{m}} \right)$ is superposition on the carrier wave of frequency ${{\omega }_{c}}$.
The resultant of the superposition wave is then passed through a square law device and a band pass filter which generates side bands from ${{\omega }_{c}}-{{\omega }_{m}}$ to ${{\omega }_{c}}+{{\omega }_{m}}$. Further, this modulated wave carries information.

Note: There are few terminologies that are used in the above solution which are square law device and band pass filter respectively.
A square law device is one that produces an output voltage or current that is proportional to the square of its input voltage or current and they are mainly used in modulators and detectors.
A band pass filter could be defined as a device that allows frequencies within a specific frequency range and rejects or attenuates frequencies that are outside the range. The low pass filter is used to isolate the signals which have frequencies higher than the cutoff frequency.