The charge of an electron was determined through a series of experiments in the late 19th and early 20th centuries. The most famous experiment was performed by Robert A. Millikan in 1909, known as the "oil drop experiment."
In the oil drop experiment, Millikan observed tiny oil droplets suspended in an electrically charged chamber. By applying an electric field to the chamber, he was able to measure the rate at which the droplets fell or rose. By carefully adjusting the electric field, Millikan was able to balance the gravitational force acting on the droplets with the electrical force. This allowed him to calculate the charge on each droplet.
Millikan observed that the charge on the droplets was always a multiple of a fundamental unit of charge. By analyzing the data from numerous droplets, he found that the smallest observed charge was consistently the same, which was determined to be the charge of a single electron. Millikan's experiments yielded an accurate value for the charge of an electron, which is approximately -1.602 x 10^-19 coulombs.
Since then, numerous experiments have confirmed the charge of an electron, including more advanced techniques such as electron beam experiments and measurements using particle accelerators. The charge of an electron is now considered a fundamental constant of nature and is a well-established value in physics.