Current

Current is the flow of electrons across a conductor.  When a voltage is applied to a conductor, the electrostatic pressure created by the voltage pushes electrons away from "negative" terminal to the positive end.  Because of naming convention current flows in the opposite direction movement of the electrons.  It is important to mention that when you flip the switch on a lamp the current is created at near the speed of light, but the electrons in the conductor move much, much, much slower.

Ampere

Amperage, also called current, is the amount of electrical energy flowing through an appliance at any given time. This measurement is expressed in units called amperes, often shortened to amps. When electricians speak of the electricity flowing in and out of your home, they may be referring to voltage, amperage or wattage depending on the circumstances. Of the three, amperage is the one you may experience first when you plug in a faulty lamp or flip the wrong switch.

Electron Drift

When electric current in a material is proportional to the voltage across it, the material is said to be "ohmic", or to obey Ohm's law. A microscopic view suggests that this proportionality comes from the fact that an applied electric field superimposes a small drift velocity on the free electrons in a metal. For ordinary currents, this drift velocity is on the order of millimeters per second in contrast to the speeds of the electrons themselves which are on the order of a million meters per second. Even the electron speeds are themselves small compared to the speed of transmission of an electrical signal down a wire, which is on the order of the speed of light, 300 million meters per second.

Definition provided by Hyperphysics