The first Zagreb index $M_1$ is one of the oldest and the most famous topological molecular structure-descriptor, defined as the sum of squares of the degrees of the vertices. In this paper we analyze and compare various upper bounds for the first Zagreb index involving the number of vertices, the number of edges and the maximum and minimum vertex degree. In addition, we propose new upper bound and correct the equality case in [M. Liu, B. Liu, extit{New sharp upper bounds for the first Zagreb index}, MATCH Commun. Math. Comput. Chem. {\bf 62} (2009) 689-698.].