In its original appearance in Judaism, "Satan" was actually a title -- haSatan, or "the Satan". (The root s-t-n in Hebrew means "adversary" or "opponent", and ha is the definite article.) It is the term for the Evil Inclination, the counterpart to the Good Inclination, which are Judaism's equivalent to the angel and devil on each shoulder (i.e., it is an internal rather than an external influence on human action). The word was also used for mortal functionaries—often what would be called investigators or spymasters today—in the courts of earthly kings. However, on some occasions it also seems to refer to that spirit or angel in God's court, who would test or question the faith of mortals. An analogy frequently used in rabbinical literature to describe this state of affairs is that of a prostitute a king hires to try to seduce his son: the prostitute, no less than the king, wants the son to pass the test and resist her advances, but is still obliged to work as hard as she can to make him fail because that's what the king wants. Sort of like the professional OPFOR (opposing force) used in militaries.
In the transition to Christianity, however, haSatan became simply "Satan" and he became the embodiment of active opposition to God. The labeling of him as an "ancient serpent" in the Book of Revelation is often interpreted to mean he was in fact the tempting serpent in the Garden of Eden. This revelation, combined with his and the demons' war against Archangel Michael and the angels in the same book, cemented his new identity as an Chaotic Evil Fallen Angel in direct opposition to God.
Popular depiction often conflates him with a line of snarky gloating about the downfall of King Nebuchadnezzar of Babylon in Isaiah 14:12 ("How art thou fallen from heaven, O Brightness (heylel/Lucifer), son of the morning!"). In addition to being mistakenly identified as Lucifer, he is also sometimes mistakenly identified as a former archangel and paragon, despite his former rank not being mentioned in The Bible canon and Archangel Michael being identified as the paragon and most powerful angel.
The earliest medieval concepts of Satan usually portrayed him as a bumbling oaf, wandering the world trying to tempt the faithful, whom the lowest of common field hands could outwit if he kept his head about him. As the Middle Ages faded into the Renaissance and beyond, Satan cleaned up his act—or, rather, had it cleaned up for him. He became an elegant, educated figure, able to mingle undetected with the aristocracy. Milton's portrayal of Satan in Paradise Lost even added a bit of true nobility to him and (unintentionally) cast him as an almost-sympathetic Anti-Hero. But he was still far less powerful than God—constrained by corporeality and bound inextricably by the promises he made, he could still be outwitted. Even so, Satan became more attractive as an antagonist. In many ways, this version of Satan is still popular today, as it provides a kind of comprehensible evil that the hero can face down and (ultimately and perhaps with help) defeat.
Since the late 19th Century Satan has been elevated by Fundamentalist Protestant Christians into a kind of Manichean Anti-God whose power is nearly on a par with that of God. (Oddly, this may be because of the rise of science and rationalism, which have essentially scuttled the idea of physical demons and devils for most people.) Almost as omnipotent and omnipresent as God, this version of Satan can be broadly impersonal or terrifyingly intimate, suiting him to Lovecraft-style horror where he can only be pushed away for a little while and never truly defeated, at least not until Armageddon itself. (And, probably not coincidentally, suiting him also to scaring the bejeezus out of and inspiring dependent paranoia in church congregations.)
A common belief (probably descended from Milton's "rule in Hell" line) is that Satan is in charge of Hell. This does not tally too readily with the orthodox Christian view, which involves Satan being eternally punished in Hell when Judgment Day comes. Could be understood as Satan simply being like the biggest and baddest convict in a maximum security prison- still punished and locked up (though The Bible does have him showing up on Earth from time to time), but still wielding Asskicking Equals Authority.
- This has happened before, as evidenced by the very availability of the word manichean to describe it, but the previous sects that thought this way were wiped out by the mainstream church and are not ancestral to the modern one.