In two dimensions a linear relationship is a relationship that can be represented graphically by a line. For instance in the relationships for momentum P=mV the momentum (P) is linearly related to the velocity V by the constant of propensity m (the mass).
A relationship that can be graphically expressed as a line in two dimensions that doesn't pass though the orgin is known as an affine relationship. An affine relationship in n dimensions can always be expressed as a linear relationship in n+1 dimensions with one of the inputs constant.
For instance P=m*V1+m*V2 is affine in an affine relationship for (m*v2 not equal to zero) between the dependent variable P and the independent variable V1 in two dimensions and in three dimensions is the dependent variable P can be thought to be linearly related to the two independent variables V1 and V2.
More technically a linear relationship is a relation that is both associative and distributive.