Why does electricity need wires to flow?
To continue to use your ball analogy think of the ball as analogous to the electron. Now what if the ball were attached to a point by a spring? Would it still fall? It can oscillate about that point but it would not be able to escape the restraining effect of the spring entirely. The same is the case with bound electrons. They are more or less bound to the atom. If the gravitational field is very strong it may be able t o break the spring and rip the ball out of the spring. This happens sometimes in electricity too. In a lightning discharge, the electric field is so high that even the bound electrons are ripped out of their atoms thus ionizing the gas and creating what is known as plasma. With a pool of free electrons and positive ions available electric current can now flow freely through the plasma - you wouldn't need wires. But unless you have a high enough electric field to produce ionization(for ionization of air the field required is close to $10^6V/m$ - such high fields cannot be produced by the 100 - 250 V household voltages available in most countries) you would have to use wires made of conducting material where free electrons are readily available if you want to have electric conduction at normal voltages.
If you define "electricity" as charge carriers in motion (which I think is reasonable), then you need free charge carriers, so you need some kind of medium from which the charge carriers can get lose.
The medium doesn't have to be metal wires, it can be gasses (as in the case of drift chambers), liquids (say a liquid argon TPC or a wet cell battery), plasma (obvious, I hope) or solids.
The atoms that make up ordinary air are not easily ionized and they recover their electrons very quickly (because of the electrostatic force). In metals man electrons are in or near the "conduction band" and can get lose quite easily and do not recombine efficiently. Electrons in the conduction band of the metal are "free" in the sense that they can move around inside the conductor easily, but it still requires energy to remove them from the metal (making them "free" in a more general sense). That energy is the "work function" you encounter in descriptions of the photoelectric effect.
The potential barrier is probably the biggest contributor to the non-flow of electricity through an open circuit.
To explain insulators in your example extend the metaphor to use cold molasses rather than water as the medium. If you are willing to wait for long enough the ball will still fall, but it will be painfully slow and you don't get appreciable useful work out of it.
Following discussions on dmckee's and Daniel's answers, here is my summary.
The electric potential in a solid metal is not 0 everywhere. It would be if electrons were localized exactly where the ion's nuclei are, so that the electric potential from electrons and the nuclei cancel exactly. However, quantum mechanics say that electrons have a wave-function that "spreads out" (delocalized). The electric potential that they create is somewhat wider than the nuclei's. The sum of both contributions, for one atom, might look like the following figure (that's a very crude and probably unrealistic view).
By the way, that allows the metals to bond together, the electrons providing the potential to attract the nearby nucleus.
If you sum these potentials from many of these atoms equally spaced, you may obtain the following potential.
Another example of such potential is depicted here. Electrons behave like waves in this potential and will bounce on each surface. That's why they do not escape easily the metal: they would have to overcome the potential wall at the metal surface. For this reason, electrons keep flowing in metallic wires.
Note that if you have electrons in excess, they will repulse each other until they are at the surface of the metal, which is why extra charge stays at the surface of conductors.