Cauterization refers to conduction of heat via a direct current from a probe heated to a very high temperature (i.e. branding cattle, grill lines on a steak)
Electrosurgery is the use of alternating current made by a generator and applied to the tissue with a handheld electrode to heat the tissue itself
When current applied in a concentrated way at a high voltage, the heat causes the fluid inside cells to vaporize and the cells to burst. Cutting mode performs vaporization which is continuous current without direct contact with the tissue. Vaporization avoids an eschar and creates a clean wound with minimal necrotic tissue at the wound borders
When current is applied over a wide surface, the heat is too low to vaporize. The tissue is heated at the surface, desiccating it and making it increasingly less conductive. The dry patch of dead tissue = coagulum.
Fulguration is the process of creating superficial coagulum by applying intermittent energy to a tissue with a monopolar device without using direct tissue contact (i.e. rollerball attachment to stop bleeding after a LEEP)
Current can also be applied with direct tissue contact, causing a deeper coagulum. Bipolar devices contain the active electrode and return electrode in their jaws. As tissue is heated, collagen and elastin in the vessel walls denature and form a hemostatic coagulum (coagulation)

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