No current is actually supposed to flow through earth when things are working properly. When we say "ground" in all our circuits we should actually be saying neutral. Neutral is where all the return current of the system flows through. Earth is a connection that is *ideally* at the same voltage as ground but carries no current (it's a dead end connection).
I'm not sure if this is entirely accurate, but I think about it this way: Neutral is at the same voltage as the earth at the POWER PLANT. And the live voltage is referenced to this voltage since it's generated at the power plant. After a long distance (ie. to your house) the voltage at the ground of your house and the power plant might be quite different...possibly dangerously different. Earth is the voltage of the ground at your house and so Earth (ground at your house) and neutral (ground at the power plant) might be quite different And the fact that current flows through neutral also makes it unsafe since, among other things, that can raise the voltage above what it's ideally supposed to be.
So, now that we've established earth is at the same voltage as the ground you are standing on when your near the plug, if you touch earth you should be fine. So what happens is earth is connected to the metal case of the equipment. Suppose a live wire (or neutral wire of high voltage) goes loose inside and touches the case. If you touched the case and it was not connected to earth the high voltage would flow through you to the ground. If the case is connected to earth the two of you form a parallel resistance connection, but since the connection of the case is a much lower resistance than you are, most of the current takes that path instead of through you (the lower resistance dominantes a parallel connection of resistors).
Think of the neutral wire as a really crappy high resistance ground wire. It should be the same voltage everywhere, but it's not so you stitch it to earth at different points. BUt for the safety reasons described above, you split it into two separate wires- one "local voltage reference" that carries no current, and one reference for the distant power supply that actually does carry current.