Saint James the Lesser is called "the Lesser" in order to distinguish him from the other James, Saint James the Great, who was the brother of of Saint John. It appears that this James got stuck with "the Lesser" in part because he was younger than the other James. Outside of a couple of mentions in the gospels, nothing is known about him or his life.
Christian tradition says that James the Lesser was martyred in Egypt; specifically, that he was crucified for having violated the Torah despite having been one of the apostles who tended to stick most closely to traditional Jewish laws. This is an implausible story because Jews sentencing someone to death would have used stoning; crucifixion was a Roman method of execution and they wouldn't have cared about some random religious person violating a Jewish religious law. There is, furthermore, no independent evidence for James being martyred anywhere for any reason.