In electricity, current is any flow of charge. Generally, a current can be associated with a flow of positive charge in one direction, or a flow of negative charge in the opposite direction. Under most circumstances, currents are caused by a flow of electrons (negatively charged), however, this is not always the case. For example, as part of chemistry, if there is salt water on one side of a membrane that lets the positive sodium ions pass, but not the negative chlorine ions, in to pure water on the other side, then a net current results. There are also instances where the electrons are the charge that is physically moving, but where it makes more sense to think of the current as the positive "holes" (the spots that should have an electron to make the conductor neutral) as being what moves. This is the case in a p-type semiconductor.See Ampere.
There is also the flow of current in the water of a river, which is where the initial idea for electrical currents is derived. In a river or stream, the water is influneced by gravity, and flows to lower areas to reduce the potential energy. This current is generally thought of as a flow of mass.
Mathematically, current is defined as the net flux through an area. Thus:
φ = j·A
where A is the area thrgough which the current is flowing, φ is the current, and j is called the "current density". The current density is defined as:
j = σnixiui i
where n is the particle density (number of particles per unit volume), u is the average velocity of of the particles in each volume, and x can be mass, charge, or any other characteristic one would like to measure the flow of.