A vector is a mathematical tool used to represent physical quantities that require both a magnitude and direction to describe. Such physical quantities include things like positions, velocities and forces.
For example, the velocity of a baseball is a vector quantity - the moving baseball has both a magnitude and a direction. The magnitude of its velocity is known as the speed, so one would say the baseball's speed is 100 km/hr. The direction can be described in many ways (once a coordinate system is chosen): the baseball could be traveling "to the left", "to the west", "in the x-direction", etc.
To learn more about vectors, visit HyperPhysics.