What is hydrostatic equilibrium and how is it used to explain the internal stability of a star?

Hydrostatic refers to fluid-like behavior.  Stars in the main-sequence are a balance between gravitational collapse and thermal expansion.  Lacking either one, the sun would either collapse under gravity to about the size of the earth, or explode like a hydrogen bomb.  Eventually, however, the sun will run out of thermonuclear fuel and become a white dwarf, weighing about a ton per cubic centimeter.

By the way, this balance is self-maintaining.  As gravity crushes a star to smaller size, this increases the temperature.  As temperature increases, the greater the force of thermal expansion.  It's like putting a weight atop a spring.

Hydrostatic equilibrium is the state of equilibrium a star experiences through most of its main sequence. The star, in effect, is matching the force of gravity, (forcing inward) and the force of the star's fusion (outward) Gravity is the giver of life and the dealer of death in a star. Gravity brings the necessary hydrogen together to create a star then, presses it so tightly that the hydrogen fuses together and alas, a star is born. The equilibrium has a huge effect on the inner stability. The force of gravity makes sustained fusion possible. Without the equilibrium, the star would blow all of it's hydrogen away from the force of fusion. When the star exists it's supply of hydrogen the equilibrium presses in on the star. In time, the star either collapses under it's gravity, causing a supernova or compresses in, forming a white dwarf.