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In the book, Introduction to electrodynamics by David J. Griffiths, he introduces potential separately as a function and potential energy through that function.

How can potential be defined before defining potential energy? Isn't the potential just potential energy per unit charge? How can potential mean something on its own?

Qmechanic
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The electric field is a conservative vector field which implies that there exists a function $V$ for which $$ \mathbf E = -\nabla V $$ We call this function $V$ the electric potential. There is no mathematical need to first define potential energy. One can then physically interpret $V$ in terms of a "potential landscape" to get intuition for what it means. In particular, positive charges "roll down" potential hills while negative charges "roll up potential hills." But from a mathematical point of view, there is no need for the intermediate definition of potential energy.

joshphysics
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    For completeness I would add the following: that the electrostatic potential energy of a charge (at rest) is defined as the work that must be done to move it from an infinite distance away to its present location, i.e., it is defined as the line integral of a certain force; that the electrostatic field is defined as a force per unit charge; and that from the previous two definitions and your definition of the electrostatic potential, it can be shown that electrostatic potential is electrostatic potential energy per unit charge. – David H Apr 25 '13 at 22:08
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    So electrostatic Potential is just a mathematical function that we assign meaning to ourselves later , however , it is just to signify that Electrostatic field is a conservative vector field ? –  Apr 26 '13 at 06:34
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    also http://physics.stackexchange.com/questions/62262/electrostatic-potential-energy-doubt ,, related to your comment . Can you explain this ? –  Apr 26 '13 at 06:36
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    @nonagon Yes I like to view electrostatic potential as a mathematical object to which we assign a physical interpretation. It is sufficient that the electric field be conservative for the existence of such a potential. – joshphysics Apr 26 '13 at 06:58
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Potential is negative of work done per unit charge by electrostatic force