What exactly is a potential? From my basic physics days I understand that potential energy is the capacity of an object to do work. 
Now I am studying fluid mechanics and I am coming across things such as "velocity potentials" and "complex potentials" that don't seem to be related to energy.
Would anyone be able to explain simply what is meant by a "potential"?
 A: In fluid dynamics, the concept of a velocity potential $\phi$  is applicable in irrotational flow where the curl of the velocity field vanishes, that is $\nabla \times \mathbf{u} = 0$. In this case, the velocity field is related to the potential by $\mathbf{u} = \nabla \phi.$
Similarly, with electromagnetism if there is no time dependent magnetic field and $\frac{\partial \mathbf{B}}{\partial t} = 0$, then one of Maxwell's equations specifies that the curl of the electric field satisfies $\nabla \times \mathbf{E} = 0$. In this case, the electric field can be specified as the gradient of a potential $\mathbf{E} = \nabla \phi$.  This is more directly related to your understanding of a potential in the context of work and conservation of energy. The energy acquired or lost by a charged particle as it moves along a path in the presence of an electric field is related to the change in electric potential which must be independent of the path and only dependent on the initial and final positions of the particle.
The unifying thread here is Helmholtz' theorem which states that  any continuously differentiable vector field that vanishes at infinity can be decomposed into  irrotational and soleniodal parts of the form
$$\mathbf{u} = \nabla \phi + \nabla \times \mathbf{a}.$$
Thus, if the velocity field is irrotational, we have
$$0 = \nabla \times \mathbf{u}  = \nabla \times \nabla \phi + \nabla \times \nabla \times \mathbf{a}$$
Since the curl of a gradient must be zero we have $\nabla \times \nabla \phi = 0$ and it follows that $ \nabla \times \nabla \times \mathbf{a} = 0$.  In this case it can be shown that $\mathbf{a} = 0$ and we have
$$\mathbf {u} = \nabla \phi$$
To further your understanding, you can read here about how the related complex potential arises in two-dimensional irrotational flow.
