How is electron charge determined?

Hint: An experiment was conducted which was used to determine the charge of the electron and further prove that all charges are multiples of charge on the electron. In this experiment, some charged oil drops were suspended against a number of forces and the known values were used to calculate electron charge.

Complete answer:
The charge on an electron was determined by an experiment which was conducted by Robert Millikan and Harvey Fletcher known as the oil drop experiment.
In the oil drop experiment, Millikan and Fletcher suspended charged drops of oil between two metal electrodes. They balanced the downward gravitational force with electric forces as well as upward drag. Since, they already knew the density of the drops; they were able to calculate the mass using the observed radii of the drops. Using the values of gravity, mass and electric field they applied mechanical equilibrium and were able to calculate the charge on oil droplets. By repeating the same experiment they were also able to determine that the charges were all multiples of fundamental value .This means that the charge is quantized.
The fundamental value came out to be $e=1.6\times {{10}^{-19}}C$.
Therefore, the charge on the electron was determined using the oil drop experiment.

Note:
Electron is the smallest charge which can exist independently. All charges are multiples of the charge of an electron as the charged bodies either have less or more number of electrons. The net forces acting on the oil drops was zero. Density is the ratio of mass and volume.