In a poly-phase system, the voltages are 360/n degrees out of phase with one another, where n is the number of phases. There is no theoretical limit to the number of phases you can have in a system - in fact some aeroplanes have 5-phase systems. However, three-phase has been settled on, and is much more convenient for two main reasons:
1) It is the most economical transmission system
2) Reversal of three-phase motors only requires swapping two phases, whereas with higher phase numbers, reversal is much harder.
As can be seen, the voltages are thus 360/3=120 degrees out of phase with each other.
In a balanced system, the currents in each phase balance out the currents in the other two. This means that there is no neutral current, and thus there is no need for a neutral conductor. The practical upshot of this is that you can transmit three times the amount of power as a single phase system does with only one extra conductor. This, as you can imagine, is a great cost saving when designing transmission and distribution systems.
I imagine that you will be predominantly dealing with unbalanced three-phase systems. These DO have a neutral current because there is a different current going down each phase and some (but not all) of the current will return along the neutral wire. Note that this occurs when the phases are together (ie, on the incoming side of the three-phase distribution board). This is something for the distribution company to worry about.
You may be wondering where the 400 volts comes from, and how you get 230v out? Basically, if you draw the phasor diagram, as below:
and measure the distance between the ends of any two phases (red-blue, blue-yellow, yellow-red), you will get 400v. This is the line voltage. There is still 230v between the end of each phase and neutral, this is the phase voltage. Note that the diagram uses the old wiring colours.
HTH,
Matt