In this paper, we explain how symmetry analysis is applied to the Korteweg-de Vries equation. This equation under study is a nonlinear partial differential equation of third order. We develop the required Lie invariance condition using local inverse theorem, an approach different from the existing Lie-Bäcklund operators and Taylor series method. This condition is then used to obtain the determining equations of the admitted Lie group. The corresponding Lie algebra is found to be solvable. We obtain the invariant surface condition, symmetry algebra and the group classification admitted by this equation. Further, we obtain the exact and similarity solutions of this equation, comment on them and represent them graphically.