This quote is from the famous "to be or not to be" soliloquy in Hamlet.
The context here is that Hamlet is thinking out loud about whether to kill himself. He decides that there is no way that anyone would want to live because life is so painful. But, on the other hand, we are afraid of what might happen in the afterlife (and this is what he is calling "conscience" here) so we are too afraid to kill ourselves.
So that is what the quote means -- that our fear of the afterlife makes us afraid to die.
What this reveals is the depth of Hamlet's indecision and the extent to which he is tortured by that indecision -- he doesn't like either of the choices that he sees for himself.
In the "to be, or not to be..." soliloquy, Hamlet is not feeling suicidal. He is merely pondering his own EXISTANCE. Killing one's self, and choosing to no longer exist are two entirely separate subjects. Hamlet, here, is merely second guessing himslef. He is not a man of action, and his over analysis of nearly every situation is his downfall. The "conscience" he speaks of here is, obviously, his own. His over thinking is making him cowardly in the act avenging his father, and murdering Claudius. While the opportunity has presented itself on a few occasions (one where Hamlet thinks Claudius is praying, and decides to wait to kill him while he is sinning), his conscience has overtaken his ability to act in a rash way. In his "turning pale", he displays his lack of courage, a characteristic which is father had plenty of.
"To be, or not to be..." refers to his thoughts of existance, where the "conscience does make cowards of us all..." part of the soliloquy moves away from pondering the afterlife, and more towards his own actions (or lack there of) and his own mental process. His conscience is preventing him from killing Claudius, not himself.