Nonsense, especially meaningless speech and often associated with spurious religious ritual.
There might be thought to be more than a hint of racial stereotyping in use of this phrase today. The casually racist belief in the gullibility of the supposedly childlike fuzzy-wuzzies was widespread when this term was coined, and it hasn't shaken that off.
The phrase probably originated from the Mandingo name Maamajomboo, which was a masked dancer that took part in religious ceremonies. In the 18th century Mumbo-jumbo referred to a West African god.
Francis Moore, in his 1738 work Travels into the inland parts of Africa noted:
"The women are kept in the greatest subjection; and the men, to render their power as compleat as possible, influence their wives to give them an unlimited obedience, by all the force of fear and terror. For this purpose the Mundingoes have a kind of image eight or nine feet high, made of the bark of trees, dressed in a long coat, and crowned with a whisp of straw. This is called a Mumbo Jumbo; and whenever the men have any dispute with the women, this is sent for to determine the contest, which is almost always done in favour of the men. One who is in the secret, conceals himself under the coat, and bringing in the image, is the oracle on these occasions. No one is allowed to come armed into his presence. When the women hear him coming, they run away and hide themselves; but if you are acquainted with the person concealed in the Mumbo Jumbo, he will send them all to come, make them sit down, and afterward either sing or dance, as he pleases; and if any refuse to come, he will send for, and whip them. Whenever any one enters into this society, they swear in the most solemn manner never to divulge the secret to any woman, or to any person that is not entered into it: and to preserve the secret inviolable, no boys are admitted under 16 years of age. The people also swear by the Mumbo Jumbo; and the oath is esteemed irrevocable. There are very few towns of any note that have not one of these objects of terror, to frighten the poor women into obedience."
That seems like a nice trick if you can do it and it isn't much of a surprise that the phrase migrated in meaning to mean nonsense and meaningless speech. In the same book Moore anticipated the eventual nonsense meaning of the term by defining it like this:
"A cant language... call'd Mumbo-Jumbo."
It wasn't until a century or so later that it began to be used generally to mean nonsense though. In 1858, The Saturday Review published this:
"The old Mumbo Jumbo of 'unchristianizing the Legislature' must not be consigned to the eternal limbo... without a parting exsufflation."
Note: exsufflation isn't a word one sees every day. With the aid of a dictionary I can say that it means 'discharge of air'.
See other reduplicated phrases.