Graph burning is a deterministic discrete-time process that models the propagation of information within a network as a set of fires that spread in a graph. The associated graph parameter, the burning number, measures the speed of that spread. The smaller the burning number of a graph is, the faster information spreads in the corresponding network. In this paper, we mainly study the burning number from a structural point of view. In particular, we structurally characterize all graphs with burning number 2. We then proceed to compute the burning number of split graphs and $\{ K_2+2K_1, P_4, C_4\}$-free graphs. In particular, we show that connected split graphs and connected $\{ K_2+2K_1, P_4, C_4\}$-free graphs are well-burnable. Lastly, we provide a blueprint on how to investigate the burning number of a given social network.