Entanglement, one of the most intriguing features of quantum theory and a main resource in quantum information science, is expected to play a crucial role also in the study of quantum phase transitions, where it is responsible for the appearance of long-range correlations. We investigate, through a microscopic calculation, the scaling properties of entanglement in spin chain systems, both near and at a quantum critical point. Our results establish a precise connection between concepts of quantum information, condensed matter physics, and quantum field theory, by showing that the behavior of critical entanglement in spin systems is analogous to that of entropy in conformal field theories. We explore some of the implications of this connection.