A scalar quantity can be repesented by a single number. Examples of scalars are temperature, salinity, volume, hydrostatic pressure, distance, time, etc.
A vector quantity represents a set of three scalar quantities or
numbers which have something in common and are collectively
referred to as a vector. Examples are velocity, force,
acceleration, gradient of a scalar, vorticity, etc.
In each case three numbers must be used to completely describe
the quantity referred to. The meanings of the numbers depend
upon the coordinates selected for describing the
particular vector. For example, if we choose to describe a
velocity 
in rectilinear coordinates, we would
then be required to specify the componnents of the velocity
for each of the three mutually
perpendicular coordinates. An illustration for the right
hand rectilinear coordinate system x,y,z is shown in
Fig. 1.
We could also represent this same vector by the values
of quantities 
representing the polar
coordinates of the end point of the vector which emanates
from the origin. In this case the term V is a scalar
indicating the magnitude of the length of the vector. There
are many other ways of representing the vector. But in every
case three components are required to determine uniquely
the vector in absolute space.
A tensor (of the second order) represents a set of three vectors or nine scalar quantities all having something in common. Elastic deformation and elastic stresses are examples of tensors.
The three stress components on each of the three mutually perpendicular faces of the prism shown above illustrate a set of nine stresses, collectively referred to as a tensor.
In effect, a scalar might be regarded as a tensor of zero
order, and a vector one of first order. This would follow
if we arbitrarily stipulate that the number of independent
scalar quantities required in order to uniquely determine the tensor
in question is 
where n is the order. Thus a tensor
of order three would be associated with 27 terms. In this
text, however, the term tensor will always be used to
denote a nine component quantity such as stress.