We investigate here the occurrence and nature of a naked singularity for the inhomogeneous gravitational collapse of Tolman-Bondi dust clouds. It is shown that the naked singularities form at the center of the collapsing cloud in a wide class of collapse models, which includes the earlier cases considered by Eardley and Smarr and Christodoulou. This class also contains self-similar as well as non-self-similar models. The structure and strength of this singularity are examined, and the question is investigated as to when a nonzero measure set of nonspacelike trajectories could be emitted from the singularity, as opposed to isolated trajectories coming out. It is seen that the weak energy condition and positivity of energy density ensures that the families of nonspacelike trajectories come out of the singularity. The curvature strength of the naked singularity is examined, which provides an important test for its physical significance. This is done in terms of the strong curvature condition, which ensures that all the volume forms must be crushed to zero size in the limit of approach to the singularity, and, also, the divergence of the Kretschmann scalar scrK=RabcdRabcd is pointed out. We show that the class considered here contains subclasses of solutions which admit strong curvature naked singularities in either of the senses stated above. The conditions are discussed for the naked singularity to be globally naked. An implication for the fundamental issue of the final fate of gravitational collapse is that naked singularities need not be considered as artifacts of geometric symmetries of space-time such as self-similarity, but arise in a wide range of gravitational collapse scenarios once the inhomogeneities in the matter distribution are taken into account. It is argued that a physical formulation for the cosmic censorship may be evolved which avoids the features above. Possibilities in this direction are suggested while indicating that the analysis presented here should be useful for any possible rigorous formulation of the cosmic censorship hypothesis.