Lissamphibia is the only surviving subclass of amphibians -- the other two, Lepospondyli (lepospondyls) and Labyrinthodontia (labyrinthodonts) having gone extinct in the Permian (290 million years ago) and the Cretaceous (120 mya) respectively. Labyrinthodonts were already going downhill by the Permian-Triassic extinction event (251 mya) and especially by the Late Triassic (210 mya), around the same time that dinosaurs became dominant. Lissamphibia consists of three orders: frogs/toads, salamanders, and the limbless caecilans.
The earliest known members of subclass Amphibia are basal temnospondyls dated to 340 million years ago, in the Early Carboniferous. These fish-like tetrapods had sprawling stances that put them close to the ground and were relatively large, 3-5 ft in length, like the lobe-finned fish they evolved from. They expended so much energy on walking that they would have been frequently exhausted and in need of rest. Thankfully for them, since carnivorous tetrapods had not yet evolved, they could rest in peace -- for a while.
The first known member of Lissamphibia, "Hotton's Ancient Frog," also known as a "frogamander," is dated to the Early Permian, 290 million years ago. As its name indicates, it has a combination of salamander and frog-like characteristics. However, even though this is known, scientists are relatively confused about the details of origin and the relationship between groups in Lissamphibia.
Every possible ancestry has been proposed -- that Lissamphibia is a clade within temnospondyls (an abundant type of labyrinthodont), a clade within lepospondyls, or that caecilans originate within lepospondyls and frogs and salamanders originate within temnospondyls. Due to the large number of shared characteristics between all members of Lissamphibia, most scientists acknowledge they are a natural clade, making the last possibility the least likely. Some scientists tend to assume that Lissamphibia evolved from temnospondyls, because they were more diverse and numerous during the time in question (the Carboniferous) than leposondyls. However, lepospondyls were considerably smaller, so it could make sense that a clade of small amphibians would evolve from another group of small amphibians, rather than from a group that is significantly larger on average.
Because of a period in the fossil record where few tetrapod fossils have been found, known as "Romer's Gap" (360-340 mya), tetrapod origins are somewhat mysterious. Recently, however, new fossil sites have been found with specimens from this elusive period. We have a lot of things to learn about this hazy period of ancient evolutionary history.
