This paper is concerned with stability analysis of biological networks modeled as discrete and finite dynamical systems. We show how to use algebraic methods based on quantifier elimination, real solution classification and discriminant varieties to detect steady states and to analyze their stability and bifurcations for discrete dynamical systems. For finite dynamical systems, methods based on Gröbner bases and triangular sets are applied to detect steady states. The feasibility of our approach is demonstrated by the analysis of stability and bifurcations of several discrete biological models using implementations of algebraic methods.