The heresy of Socinianism is named after its founder, Faustus Socinus (1539-1604) and gained popularity among the Polish Brethren and the Unitarian Church of Transylvania. Socinians had a number of heretical beliefs, including rejecting substitionary atonement, adopting and embracing Pelagianism, Arianism, and Open Theism. However, none of those heresies are unique to Socinianism.
Socinianism, as the term is used, refers to a denial of the miraculous in lieu of a dependence upon the scientific or philosophical. Socinianism is a spin on skepticism. Skepticism is a philosophy that typically rejects the miraculous or supernatural in lieu of atheism or agnosticism. Socinianism is skepticism that seeks to retain the notion of professed Christianity; in other words, it is a skeptical Christianity that denies many clear claims of Scripture.
Modern Socinians include Unitarians and Christodelphians, but also include groups like Bio Logos that deny the more supernaturalistic claims of Scripture in subjection to modern scientific theory and have renown modern proponents, like Tim Keller of Redeemer Bible Church.