The main sequence of events of fertilization in higher animals especially vertebrates is best understood on the description based on what is observed in the sea urchin because the egg of such animals are easily collected and fertilization takes place externally in sea water where it can be studied and observed under the microscope.
The spermatozoon comes in contact with the egg by random movement. When the head of a spermatozoon hits the vitelline membrane, the acrosome at the tip of the head bursts releasing a substance which softens the vitelline membrane at the point of contact as a result of the membrane lining the acrosome that forms a thin filament which pierce the egg membrane. The whole process is called the acrosome reaction and this helps the sperm cell to penetrate into the cytoplasm of  the egg.
A rapid chemical change at the surface of the egg prevents any further entrance of any other sperm. A viscible sign is the thickening of the vitelline membrane. This results from the cortical granules migrating through the plasma membrane and applying themselves to the inner surface of the vitelline membrane and there is accumulation of fluid immediately below the newly formed thickened vitelline membrane with the result that the vitelline membrane appears to lift off from the egg surface to form a fertilization membrane. This serves as a barrier to the entry of other spermatozoon.
After a spermatozoon has penetrated the egg, the tail is detached while the head and middle piece are pushed through the cytoplasm toward the egg nucleus.