In a solid metal conductor, current is the flow of free electrons (negatively charged) which move from negative to positive. Creating an overall movement of charge in this direction.
In molten or dissolved electrolytes, it is the flow of negative ions (anions) towards the positive terminal and vice versa. For more about this see electrolysis in the chemistry section
There are two types of current. Alternating current (AC) and direct current (DC).
Direct passes in the same direction constantly. This is used to power electrified railway lines such as the London Underground.
Alternating current on the other hand, is always changing direction. This is the form that domestic electricity supply takes.