Why do we define the stability of a body only under the action of conservative forces? Why do we analyse the $stability$ (Stable, unstable, or neutral) of a body only under the influence of a conservative force? What if non-conservative forces are acting on the body as well? 
For example, a marble rolling around in a round-bottomed bowl tends to move towards the bottom surface of the bowl and when it is at the bottom, and not moving, it is in stable equilibrium. While the same marble kept on top of an inverted bowl is in unstable equilibrium. 
Although I understand mathematically, and physically, why the marble is in stable equilibrium in the first case and why it is in unstable equilibrium in the 2nd case, what's confusing to me is that my textbook says that we can analyse the state of equilibrium of a body that under the influence of a conservative force. But in these two cases, there is Normal reaction force acting on the marble too, which is a non-conservative force. 
Likewise, in this case, there is a Normal force acting on the feet of these acrobats. 

What I am trying to understand is, why does my textbook explain the state of equilibrium under the action of a conservative force when there are non-conservative forces acting too?
 A: 
For example, a marble rolling around in a round-bottomed bowl tends to
  move towards the bottom surface of the bowl and when it is at the
  bottom, and not moving, it is in stable equilibrium.

That is correct. A system is in stable equilibrium if when displaced from equilibrium it experiences a net force or torque in a direction opposite to the direction of displacement. For your marble example, that would be the marble resting at the bottom of the bowl. If displaced (pushed) towards one side of the bottom of the ball and released, it will experience a force opposite the displacement ultimately restoring it back to its initial (equilibrium) position at the bottom of the bowl. 

While the same marble kept on top of an inverted bowl is in unstable
  equilibrium.

Correct. A system is in unstable equilibrium if it accelerates away from its initial equilibrium position if displaced even slightly.

what's confusing to me is that my textbook says that we can analyse
  the state of equilibrium of a body that under the influence of a
  conservative force. But in these two cases, there is Normal reaction
  force acting on the marble too, which is a non-conservative force.

Although both cases involve normal (non-conservative) forces, I believe the normal forces only relate to equilibrium (in the vertical direction) and are not related to whether the equilibrium is stable or unstable. For the marble at the bottom of the bowl (stable equilibrium), a displacement temporarily increases gravitational potential energy but there is a restoring conservative force (gravity) to return it to its original equilibrium state. At the top of the bowl (unstable equilibrium), no matter which way the marble is displaced, gravitational potential energy continues to go down until a new equilibrium state is reached.

Likewise, in this case, there is a Normal force acting on the feet of
  these acrobats.

Yes, but the normal force only provides equilibrium, but not stability, as in the case of the marble and bowl.
Hope this helps.
