Electrostatic phenomena result from the ability of electrons to move relatively easily in many materials. There are two general classes of electrostatics: induction and conduction.
Induction is usually a temporary state where electrons within one substance are attracted or repelled by a nearby charged object. For example, if a positively charged object is brought near a neutral object, the electrons in the neutral object are attracted to the positive charge and the result is that within the neutral object the side toward the positive charge develops a slightly negative charge and the side opposite the positive charge develops a slighhtly positive charge. The net charge of the neutral object is still zero but the electrons are no longer evenly distributed within the object. When you remove the positive charge the electrons within the neutral object redistribute and go back to their original state.
When conduction occurs, a charged object is brought into actual contact with a neutral object. During this contact, electrons actually move between the two objects and when the objects are separated both objects now have the same charge.
For example, a negative charge is brought into contact with a neutral object. Some of the excess electrons will move from the charged object to the neutral object. When they separate the neutral object now has a negative charge (gained electrons) and the negative object is still negative but less so (lost some of its excess electrons). For the case of a positive object and a neutral object brought into contact electrons now move from the neutral object to the positive object and both become positively charged.
An electroscope works on the same principle and is used to detect charge in an object.
Some practical applications of electrostatics are powder spray painting, electrostatic air filters in homes, and electrostatic filters on tops of smokestacks of power plants burning coal.
Electrostatic charges also exert forces on each other which can be calculated using Coulomb's Law: F = k Q1Q2/d squared.
Electrostatic phenomena are quite commonplace, but unusual events. There are many different examples of such events, but I will name only a few of them to demonstrate what they are:
One example is lightning, an electrostatic discharge produced during a thunderstorm. There are several hypothesis that attempt to explain this phenomena, and the electrostatic induction hypothesis says that a charge separation occurs as the result of strong updrafts which carry water droplets high enough to become supercooled. These droplets then collide with ice crystals creating an electric field because of the distribution of positive and negative charges.
Another example is an electrostatic discharge which results from an imbalanced charged in an object that produces a visible attraction, repulsion and sparks. It most often occurs when two materials come into contact and then are separated. The friction between the two materials tribocharges the atoms that creates an difference in electrical potential.
In all cases essentially what happens with the electrons is that they group together to form a negative electrical field resulting from the separation of the positively charged ions and the negatively charged ions during an electrostatic event.