This paper considers general finite-dimensional estimation algebras associated with nonlinear filtering systems. General considerations and approaches toward the classification of finite dimensional estimation algebras are proposed. Some structural results are obtained. The properties of Euler operator and the solutions to an underdetermined partial differential equation, which inevitably arise in an estimation algebra, are studied. These tools and techniques are applied to the study of finite-dimensional estimation algebras with state dimension 2 to obtain a complete classification result. It is shown that a finite-dimensional estimation algebra with state dimension 2 can only have dimension less than or equal to 6. Moreover, the Mitter conjecture and the Levine conjecture hold for finite-dimensional estimation algebras with state dimension 2.