Effective mass of electron (hole)

For a given dispersion relation $epsilon (vec k)$ for electrons in a crystal , the tensor of the reciprocal effective mass is defined by $$ (1/m)_{ij}= frac{1}{hbar^2} frac {partial^2 epsilon}{partial k_i partial k_j}$$ where $k_i$, $k_j$ are the components of the wave vector $vec k$. When you chose the principal axes of this tensor for the $k_i$, you get an infinite mass $m_i to infty$ in a considered direction $k_i$ when $$frac{1}{hbar^2} frac {partial^2 epsilon}{partial k_i^2}=0$$ which happens at inflection points of the dispersion function in this direction, and $m_i=0$ when $$frac{1}{hbar^2} frac {partial^2 epsilon}{partial k_i^2} to infty$$

@freecharly has given an excellent answer! In human words it means that the effective mass is the curvature of the conduction/valence band near its minimum/maximum (respectively for electrons/holes).

It is worth adding that electrons and holes in a crystal lattice are not free particles, but excitation of a complex system. Effective mass is just a way of making them resemble free particles by expending their energy to the second order in quasi-momentum. In other words, physically it is not the same as the real mass.