For current to flow, you must have a closed circuit. This means a loop that starts at one transformer terminal, goes through your load, and goes back to another transformer terminal. A single closed circuit is the minimum required for current flow (ignoring capacitance).
A neutral need not be present, the only requirement is a closed circuit.
A neutral is transformer terminal that is at the vector zero voltage of a set of transformer terminals. It could be the center tap of a single phase secondary, or it could be the common tap of several coils in a wye connected transformer or bank, or any of a number of other possibilities. A neutral conductor is connected to this transformer terminal, and may be part of a circuit carrying current. But the neutral, even if present, is not necessarily used.
One of the transformer terminals may be connected to 'earth' in order to create a grounded system. This terminal is connected to the 'grounded conductor'. A grounded conductor is not required, but is almost universal for electrical supply. The grounded conductor need not be the neutral, although if a service supplies a neutral, code requires that the neutral be used as the grounded conductor. If the service does not supply a neutral, then one of the other circuit conductors may serve as the grounded conductor.
-Jon