For a connected graph $G$, its resistance distance matrix is denoted by $R(G)$. A graph is called resistance regular if all the row (or column) sums of $R(G)$ are equal. We provide a necessary and sufficient condition for a simple connected graph to be resistance regular. Additionally, we establish sharp bounds for the resistance spectral radius and present various bounds for the resistance energy of $G$. Furthermore, we compute the resistance spectrum and resistance energy of some resistance regular graphs.