A "basal" vertebrate means a vertebrate at the base of the vertebrate family tree, either one of the vertebrates from which all others evolved or a close cousin of the same. Some basal vertebrates are stem group vertebrates, meaning they are older than the earliest common ancestor of all living vertebrate species.
Vertebrates, like all other animal phyla, are extremely old. The first basal vertebrate fossil is likely Yunnanozoon lividum, which lived 530 million years ago in what is now China, although some researchers disagree, calling the species a hemichordate rather than a real chordate (vertebrate). Another basal vertebrate, Pikaia gracilens, resembles a modern lancelet. A third basal vertebrate, Myllokunmingia, is a primitive fish and one of the earliest vertebrates. Depending on the article, web page, or paper, each is sometimes referred to as the oldest vertebrate.
It is a mystery as to how basal vertebrates initially evolved. There is a general consensus that a spinal column is helpful for swimming, giving muscles something to pull against, producing the sleek "S" shaped locomotion we associate commonly with fish. The first vertebrate may have been a sediment-dwelling animal that evolved a flatter body for swimming, a mobile larva of some sort that began retaining its larval features into adulthood, or a stationary filter-feeder that evolved into a free-swimming animal. We don't know for sure, because the fossil record is far from perfect.
According to molecular studies, tunicates, also known as sea squirts, are the closest living relatives of vertebrates. Tunicates are considered chordates, but not vertebrates proper, because they have a notochord during the early stage of their development. Other relatives of the vertebrates include the hemichordates, which includes animals like the arrow worm, and the echinoderms, which includes starfish and relatives.
Whatever the cause of the evolution of basal vertebrates, they became a very successful group relatively shortly. From the Paleozoic onwards, fish, the most numerous vertebrates, have largely ruled the seas. For evolving large and sophisticated land animals, a backbone is a must. Invertebrate land animals, like snails and insects, don't have the tendency to grow very large or evolve much of a complex brain.