We present here the first classification of pencils of quadrics based on the type of their intersection in real projective space and we show how this classification can be used to compute efficiently the type of the real intersection. This classification is at the core of the design of the algorithms, presented in Part~III, for computing, in all cases of singular intersection, a near-optimal parameterization with polynomial functions, that is a parameterization in projective space whose coordinates functions are polynomial and such that the number of distinct square roots appearing in the coefficients is at most one away from the minimum.