In its original appearance in the old Israelite faith tradition, "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.) "haSatan" exclusively refers to the figure known as Satan, an angel who God commands to spread temptation among the human race, though in later tradition the being, and by extension the term, was broadened to refer to "yetzer hara", Evil Inclination, the counterpart to the Good Inclination, essentially making up 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). Without the prefix of "ha", the word "satan" was used to describe normal, human enemies, and 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. In modern times, a more familiar analogy would be the professional OPFOR (opposing force) used in militaries, or — from Willy Wonka & the Chocolate Factory — the actor hired to pose as Wonka's business rival Slugworth and test the kids' loyalty to Wonka.
However, autonomous and unambiguously evil spirit powers are also acknowledged in the Old Testament; for example, the gods of the heathen nations are said to be demons, some of them quite powerful; there also exists a "king of terrors" who lives in Hell and feeds on the souls of the dead. The prophets speak of a fallen cosmic ruler cast down in Hell, who was once a cherub, or angel of the highest order (though other interpreters think this is mere hyperbole describing a human king), and various books speak of a monstrous cosmic dragon (a sort of Jormungandr prototype) as God's most powerful enemy. All of these are traits which Christian theology ascribe to Satan, though in Old Testament times they were not yet explicitly linked to the name of ha-Satan. All of these have roots in the broader religious landscape of early Judaism; some scholars have conjectured that Judaism may have shifted from polytheism to monotheismnote under the influence of Zoroastrianism (which thanks to Darius the first really left a good impression on early Jews), so rival deities from other people had to be rationalised as evil. Zoroastrianism itself has the first recorded example of a manifestation of pure evil, Angra Mainyu, which is in some ways a prototype for what Satan would latter become.
In late Jewish tradition, such as the Dead Sea Scrolls and the Martyrdom of Isaiah, all of these intimations were interpreted as describing one entity, and ha-Satan became simply Satan, the universal Adversary, as he came to be understood as the personal embodiment of active opposition to God. The Wisdom of Solomon (a late Jewish book, considered canonical by Roman Catholics and Eastern Orthodox Christians) describes him as responsible for the original Fall, while the Book of Revelation calls him "the ancient serpent," a name often interpreted to mean that 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. It's worth noting however that Genesis itself gives no indication the serpent was anything other than a serpent.
The later texts of the New Testament presents Satan as an incredibly powerful and absolutely malevolent Cosmic Entity far beyond human ability to fathom. While not a true Anti-God in the dualistic sense, he is portrayed in terms that suggest god-like power. He is said to control both the micro- and the macrocosmos, from the "elements of this world" to the "celestial bodies" and beyond; his lieutenant demons are called "rulers of the universe" (Kosmokratores); himself, he is frequently referred to as "the Lord of this world" and once, actually "the god of this world"; and indeed, it is written explicitly that "the whole world is under the control of the evil one." Further, the physical form he affects when making a personal appearance is beautiful and terrible as an "angel of light"; mortals (even devout Christians) spontaneously react to the sight of his lesser creatures, such as the Beast of Revelation and the Whore of Babylon, with wonder and almost worshipful admiration. Finally, he rules millions and millions of very powerful fallen angels, and can easily assume direct control of anyone in the world who is evil (or even just sufficiently non-Christian, really), and/or imbue them with the powers of a minor Physical God if he so chooses. Described in pop-culture terms, he is probably best understood as a very powerful Dimension Lord or God of Evil: In most any pagan pantheon, he would probably be the God of Gods. Fortunately for humanity, he lacks the absolute omnipotence and omniscience of the one true God, which is why the final victory of good over evil is still assured, in spite of his cosmic near-supremacy (though even without God, it's noted that he only had one-third of angels behind him, and Archangel Michael's forces could defeat his alone).
The wider pop-culture and folklore conception of Satan is often rather different from the Biblical picture, tending, perhaps for dramatic reasons, to make him a much less formidable figure that mortal agencies can realistically combat and thwart. This version, too, rests on a long, though younger tradition. Medieval folktales often portrayed him as little more than 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. (Echoes of such humorously pathetic demons still abound; for example, in many of the works of C. S. Lewis, though then of course in a much more sophisticated form.) As the Middle Ages faded into the Renaissance and beyond, Satan cleaned up his act somewhat — 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.
In several influential strains of modern Christian thought, Satan's immense cosmic agency and power are again emphasized, if they were ever really forgotten, and some of this has also bled through into much contemporary fiction. Nearly as omnipotent, omniscient and omnipresent as God, this old-new 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.