The common ancestor of all snakes is indeed a lizard. So, technically, all snakes are descendants of a legless lizard. Snakes used to be classified in a different order than the lizards, but now practically all scientists have reclassified snakes in the same order as lizards. The new order is known as Squamata. However, snakes are classified in its own suborder, separate from all other lizards. That is because even though leglessness has evolved many different times among the lizards, none of these legless lizards has evolved to the same extent as snakes and none of the legless lizards is particularly close to the snakes phylogenetically.
Although there is some disagreement, many scientists agree with Dr. Lee and Dr. Caldwell that snakes are most closely related to the monitor lizards. Snakes probably evolved in a marine environment, from a marine lizard closely related to the extinct mosasaurs, which are lizards classified in the same family as the living monitor lizards. Because of that, Dr. Michael Lee has found similarities between the eyes of snakes and the eyes of fish. These similarities are due to convergent evolution, or similar adaptations to a similar (in this case aquatic) environment. Most legless lizards, however, still retain eyelids and they evolved on land. Snakes are also quite elongated, much longer than most lizards. Legless lizards also tend to retain at least some of the pelvic bones, but a vast majority of snakes have lost all traces of the pelvic bone. Snakes have also evolved a big mouth that allows most of them to feed on prey that no lizard can handle. The oldest known fossil snakes have this feature, even though these snake fossils still retain tiny hindlimbs. These snakes were also found in marine deposits. Although some snakes have evolved smaller, less mobile jaws, these appear to have evolved as adaptation to insect feeding in underground habitats, in which there is little room to expand the jaws and also little need since insects are easy enough to swallow with small jaws.
Based on these anatomical differences and others, snakes were originally classified as a separate order from the lizards, until more recently available evidence shows that snakes actually are more closely related to lizards than to any other reptilian order, so close in fact that they were combined into a single order. Despite their close relationship, snakes branched off from their lizard ancestor before the extinction of the dinosaurs, and they have evolved a good deal of new features not found in any legless lizards. For these reasons, snakes are placed in a suborder of their own, and not regarded as just legless lizards.