The evolution of cooperation among unrelated individuals is studied in a lattice-structured habitat, where individuals interact locally only with their neighbors. The initial population includes Tit-for-Tat (abbreviated as TFT, indicating a cooperative strategy) and All Defect (AD, a selfish strategy) distributed randomly over the lattice points. Each individual plays the iterated Prisoner's Dilemma game with its nearest neighbors, and its total pay-off determines its instantaneous mortality. After the death of an individual, the site is replaced immediately by a copy of a randomly chosen neighbor. Mathematical analyses based on mean-field approximation, pair approximation, and computer simulation are applied. Models on one and two-dimensional regular square lattices are examined and compared with the complete mixing model. Results are: (1) In the one-dimensional model, TFT players come to form tight clusters. As the probability of iteration w increases, TFTs become more likely to spread. The condition for TFT to increase is predicted accurately by pair approximation but not by mean-field approximation. (2) If w is sufficiently large, TFT can invade and spread in an AD population, which is impossible in the complete mixing model where AD is always ESS. This is also confirmed by the invasion probability analysis. (3) The two-dimensional lattice model behaves somewhat in between the one-dimensional model and the complete mixing model. (4) The spatial structure modifies the condition for the evolution of cooperation in two different ways: it facilitates the evolution of cooperation due to spontaneously formed positive correlation between neighbors, but it also inhibits cooperation because of the advantage of being spiteful by killing neighbors and then replacing them.