There are two kinds of "earth" being talked about here. There is:
- The kind that aims to use the ground itself as a "return path"; and
- A protective "earth", which is actually a separate conducting line laid throughout a building.
For the "return path" earth of 1. there are several ways wherein this will work:
There is actual conduction (i.e. drift of charge like ions) through the water and dissolved ions in the ground and this effect is sometimes (albeit seldom) used to lessen the amount of infrastructure one needs in remote areas to deliver mains power to sparsely spread buildings. As you can imagine, its efficiency depends greatly on groundwater levels and so this is far from ideal. See the Wiki page on Single Wire earth Return for details.
There is "displacement current return path" as described by Kornut's Answer, which I'll add a little more to. As I talk about in detail here, one actually doesn't need a return path to make an electric "circuit" work. We can put some numbers in to flesh out Kornut's Answer: imagine that your "earth" connexions are instead big conductive spheres that store charge as in my drawing below.

You can easily do the Gauss's law calculations for widely separated spheres to show that the voltage difference between them when one has been "pumped up" with positive charge $+Q$ and the other with negative charge $-Q$ in the way described by Kornut in his answer is:
$$V = \frac{Q}{2\,\pi\,\epsilon_0\,R}$$
where $R$ is the spheres' radius (assumed the same), so that the above electrically looks like a "circuit" with the spheres replaced by a capacitor of value $2\,\pi\,\epsilon_0\,R$. So, at $50\mathrm{Hz}$, say, with one metre radius spheres, the impedance $(i\,\omega\,C)^{-1}$ will have a magnitude of $(100\,\pi\,\epsilon_0)^{-1} = 360\mathrm{M\Omega}$ so you're going to need awfully big spheres or high frequencies to make the above scheme light the lightbuld. Even at $1000\mathrm{km}$ radius, the figure is $360\Omega$.
There have been some really bizarre truly one-line power transmission systems thought of in the past, where the one line works as a waveguide and does not need a return path. See the Wiki Page for the Goubau Line for details.
Now to discuss the notion of protective earth. In home electricity supplies, one side of the supply ("the neutral") is "tethered" to the same potential as a protective earth circuit. This latter is simply a system of conductors, going through the third "Earth" pin on the socket outlet, that tethers any conducting surface of an electrical appliance to the one side of the supply. The other side of the supply is called the "active". If a fault happens in an appliance such that the active touches the conductive housing of an appliance (say of a toaster), we have a dangerous situation, since anyone touching the appliance can then get an electric shock. However, if the housing is connected to the protective earth circuit, there is a redundant path back to the supply's neutral. This leads to a high current in the redundant path and hopefully a blown fuse (the redundant protective earth path has a fuse in it that blows with a very low current).
A more modern and safer way to achieve this protection is earth leakage protection, which I talk about in this answer here.