Electrical current's role in a processor?
Simply put, current is used to charge capacitors to a given voltage.  It is then this voltage that is representative of a "1" or a "0".  The voltage is the state, the current is the juice that gets you to that state.  THis applies to CMOS processors, but since there are few other kinds around this is probably general enough.  I'll let the pedants worry about other kinds.

On Edit, I thought I'd mention that the capacitances being charged are the internal gate capacitances of the transistors themselves.