What is the song "Sympathy for the Devil" by the Rolling Stones about?

In spite of common belief, the song is not seriously arguing Christian doctrine, but using the devil as a symbol, a personification of evil, and of how it works in the world.

If you take the lyrics at face value, you might consider that Mick Jagger (as lead vocalist and lyricist) is sincerely suggesting that the devil really isn't such a bad fellow. After all, he says, “every cop is a criminal, and all the sinners saints,” and it was “you and me” who “killed the Kennedys.” So what's the difference? We're all the same, aren't we? He's just one of the lads.

Look more deeply, though, and you will find that this confusion of good and evil, of appearance and reality, is really just a ploy of the evil one. The devil may appear to be a gentleman, “a man of wealth and taste.” He may appear to be polite and a member of good society, approaching you with the words, “Let me please introduce myself,” and “pleased to meet you.” But alongside of this we have a long catalog of some of the worst atrocities of human history, starting with the torture of Christ on the cross, and ending with the modern assassination of the Kennedys (so contemporary, in fact, that the lyrics were changed from “John Kennedy” to “the Kennedys” while the Stones were in the studio recording the song). Along the way we have the Crusades and the Bolshevik revolution in Russia thrown in for good measure. So let there be no mistake: there really is evil in the world.

This, then, is the real significance of the taunting refrain, “Hope you guess my name, But what's puzzling you is the nature of my game.” This is not Mick Jagger, saying “hope you can guess that I'm playing the devil in this song.” That's obvious from the beginning. No, this is the devil saying, “Hope you recognize me when you see me, because I come in many guises. And my game is not to do evil myself, but to trick you into doing it.”

This confusion between appearance and reality, between good and evil, runs throughout the song. Christ is mentioned, not only for the pain that he experienced, but for his “moment of doubt.” And in addition to all the acts of violence described, the devil says that he has “stolen many a man's soul and faith.” His interest in the son of God was not limited to the torture of Christ himself — he also “made damn sure that Pilate washed his hands and sealed his fate.” And his final threat to the listener is to “... lay your soul to waste.”

Just in case we might be in danger of taking Jagger's appropriation of the devil as a serious case of religious conviction, he throws in the lines, “I watched with glee, while your kings and queens fought for ten decades for the gods they made.” So Jagger is not seriously arguing Christian doctrine here, but using the devil as a symbol, a personification of evil, and of how it works in the world.

When attempting to interpret the song, we should be cautious about recognizing the devil as a literal figure. The Stones here are attempting to explore the duality of human nature. Many times we are blessed with fortune, happiness, and life (good) and at other times we are cursed with destruction, pain, and death (evil). Consequently, human beings have established social institutions (religions) to explain the nature/existence of good and bad. Judeo-Christian religions have generally sought to explain the nature of evil through the figure of (pleased to meet you, hope you guessed his name) Satan.

In general, the narrator of the song can be seen as a biblical Satan who manifests himself through greed, malice, and misunderstanding. More specifically, he represents the negative aspects of human nature. Here, however, we encounter a paradox (what's puzzling you); humans would have no concept of "evil" if there was nothing good to compare to.

Jesus Christ taught love, compassion, and tolerance (good), and was brought before Pilate, who refused to intervene on his behalf. Pilate could be seen as evil, but without his serious error in judgement, Christ would not have died for the sins of the world.
As christianity began to spread widely throughout Europe, medieval rulers and early popes soon established close relationships to legitimize each other's authority and thus mutually expand power. Theocracies here were not uncommon because ecclesiastical law contributed greatly to the power of nobles and popes. Pope Urban II spawned the first crusade by drawing attention to the inherent differences in culture between Islam (Infidels = evil) and Christianity(good).
Czar Nicholas II represented the last of the feudal monarchs, and was overthrown in order to abolish the inhumane and outdated custom of binding peasants to the land. While some equality was achieved (good), the entire noble family was slain, and bloody revolutions, wars, and dictatorships have permeated the remainder of Russian history.
The evils and atrocities commited by the Nazi regime need no clarification here, yet the oppressed German people saw Hitler as a father-like figure who would restore prosperity to Germany and relieve the heavy burdens of the Versailles Treaty. Consequently, we cannot blame Rommel because he "drove a tank, held a general's rank" for the percieved benefit of the German people
Who killed Robert and John F.? After all, we did.
In comparison, "...every cop is a criminal, and all the sinners are saints." Evil is an inherent part of every human conscience; along with good, it is part of "the nature of [our] game." The line "as heads is tails, just call me Lucifer," creates further irony. Lucifer, literally translated, means "morning star," which was the name of Satan before he fell from heaven.

Lastly, it is no coincidence that this song was written in the first (I/me), and not the third person (him/he). Not even troubadours (musicians, i.e. the Rolling Stones) are immune to evil. Nor are the wealthy and tasteful. The last lines and song title, in fact, leave us with an ultimatum to have some "Sympathy for the Devil." Essentially, this means that when we encounter pain, fear, and sadness, we should not simply chalk it up to a synthesized literary figure with horns, but instead we shuold seek to better understand both the good and the bad in ourselves and in others.