What do external forces have to do with equilibrium?
Equilibrium refers to a state in which all of the external forces on an object all balance each other such that there is no net effect (net force equals 0). An object in equilibrium will not experience acceleration, and will either remain at rest, or continue moving at a constant velocity.
Here's a simple example, pick up a book (or any random object) and hold it up in the air. The book is now in equilibrium, the downwards force of gravity is perfectly countered by the upwards force that you are applying to it. Notice that the object neither falls nor goes upwards (i.e. no acceleration).
Now let go of the book, notice how it falls downwards till it hits the ground (or whatever was beneath it). That is because without the upwards force applied by your hand, the object is no longer in equilibrium, and the force of gravity takes over until it is in equilibrium again.
Loren Chang, Physics Undergrad. Student, UC Irvine
The nature of this question indicates you are just being introduced to the idea of equilibrium. The term 'external force' is apparently adding some confusion. Let us just think very simply about the word 'equilibrium'. What does it mean to you, what does it imply? I offer that it implies things are 'balanced', such that all the forces applied add up to ZERO! That is equilibrium.
Now, the whole concept cannot be discussed without reference to something, some particle, some body, some system .. something specific. For the purpose of illustrating this question, let's say the system is YOU standing in a room. You are in equilibrium, you are not moving, you push on the floor with your weight, the floor pushes back. Now, someone comes along and pushes down on you! There is an addition of an external force ... and you still stand there .. you push the floor harder .. and it pushes back harder .. but equilibrium is maintained. And the case we are discussing is STATIC ... that is .. most ordinarily speaking .. not moving.
The DYNAMIC situation is similar, but a bit more complicated. Let's use the same example, but now the person comes along and pushes you sideways enough to make you fall, or slide, or move somehow. It is an external force applied to the original system, but now you move .. you 'accelerate'. Well, everything is still in equilibrium believe it or not! But the dynamic equilibrium now includes more difficult forces to know much about .. inertia forces .. the fact the you have mass and were accelerated .. you develop an inertia force resisting the push. All forces are still in balance.
External forces are arbitrary to some extent .. in that, as an analyst, YOU choose the system dimension and keep track of what is 'external' and what is 'internal'. Before you walked into the experiment room to stand the floor was in equilibrium. When you walked in YOU were the external force .. and the floor did not cave in .. it was in equilibrium .. its weight was supported by the beams in below it fastened to the walls .. and it still is in equilibrium when you stand there.
I trust you begin to get the idea. It depends on how much of the world you include in your system .. but be assured .. equilibrium prevails. I hope this simplified explanation was what you were seeking.
Bruce Miers, M.S., Practicing Engineer - Retired