Using an all-electron, full potential first-principles method, we have investigated the topology of charge density and elastic properties of the two polymorphs, alpha and beta, of Ti3SiC2. The bonding effect was analyzed based on Bader's quantum theory of "atoms in molecules" (AIM). It was found that the Ti-Si bonding effect is significantly weaker in beta than in alpha, giving less stabilizing effect for beta. The Si-C bonds, which are absent in alpha, are formed in beta and provide additional stabilizing effect for beta. In contrast to conventional thinking, there is no direction interaction between Ti atoms in both alpha and beta. The calculated elastic properties are in good agreement with the experimental results, giving the bulk modulus of about 180 GPa and the Poisson's ratio of 0.2. The beta phase is generally softer than the alpha phase. As revealed by the direction dependent Young's modulus, there is only slight elastic anisotropy in Ti3SiC2. For alpha, Young's modulus is minimum in the c direction and maximum in the directions 42o from c. For beta, the maximum lies in the c direction, in part due to the formation of Si-C bonds in this direction.