The phrase the tail wags the dog means:
– an unimportant or subsidiary factor, person or thing dominates the situation;
– the usual roles are reversed.

It is based on the image of the inversion of the natural order.

The earliest instances of this phrase that I have found are from The Brooklyn Daily Eagle (Brooklyn, New York) of Tuesday 6^th December 1870:

A Tail that Wags the Dog.

A muddy-headed man writing on finance, is nearly as deplorable an object as a man who cannot swim struggling in water beyond his depth. The President, in his message, thus delivers himself upon the tax and tariff questions:
By steadiness in our present course there is no reason why, in a few short years, the national tax-gatherer may not disappear from the door of the citizens almost entirely. With the revenue stamp dispensed by postmasters in every community, a tax upon liquors of all sorts, and tobacco in all its forms, and by a wise adjustment of the tariff which will put a duty only upon those articles which we could dispense with, known as luxuries, and on those which we use more of than we produce, revenue enough may be raised after a few years of peace and consequent reduction of indebtedness to fulfill all our obligations.
There are several funny things in this passage. One is the closing inference that we are not now “fulfilling our obligations”; that it will take “a few years of peace” yet to make the receipts cover the expenses. And then that vision of “the national tax-gatherer disappearing from the door of the citizen almost entirely in a few short years,” is quite radiant. There is still to be such a functionary, and he is still to hover around our door; but very stealthily, so that we shall barely see the disappearing coat tails turning the next corner, as we glance out from our stoop upon his retreating figure.
There are a good many qualifications suggested, even to the partial relief we are to experience from the almost entire disappearance of the national tax-gatherer from the immediate vicinity of the house door. He is still to haunt the whisky distillers and the tobacconists, in all forms. The marines need not flatter themselves that the Commander-in-Chief sees, even in the vista of years ahead, any prospect of releasing them from the pre-election duty of raiding [cf. note 2] in the Fifth Ward. The Whisky Ring [cf. note 1], in and out of office, need not fear that their little game is likely to be played out, even with the lapse of years. Those taxes that pay the officials best for “seeing” the revenue evaders, are still to be retained. And the stamps are to be kept in vogue, but transferred wholly to the post office. There every man who wants to sell a box of matches, worth intrinsically less than one-half of a cent, will have to go, and pay one cent for a stamp, take another cent’s worth of trouble to stick it on, and then sell the box for three cents, instead of the one cent it would cost but for the stamp. None of the vexatious little stamp taxes, which worry the business man, are to be removed, any more than the great whisky tax which enriches the dishonest official.
But there is to be a great boon in the tariff schedule, if not in the tax list. The President wants “a wise adjustment,” which will “put a duty only upon luxuries, and on those articles which we use more of than we produce.” Here’s richness. Nothing on earth is to be burdened with a tariff, only all luxuries, and all necessaries of life except our great agricultural staples—for of all else we “use more than we produce.” To-day, we “use more than we produce,” of every article which we import into the country. If not, we would not import it. Well, then, the President would have a duty laid, only on all things whatever, that are imported, and on all luxuries—nearly all of which are imported—besides.
It may not be unfortunate that we need a tariff for revenue. It may not be unfortunate that we have President Grant [cf. note 3] as our Executive. Yet assuredly it is most unfortunate that we have these two coincidently. For, obviously, President Grant does not know what he is writing about when he writes about the tariff, whose wise adjustment seems to him to require duties, only on luxuries, and on all imports whatever. The exceptions more than cover the whole scope of the rule; the tail wags the dog. The anatomy of the tariff is a science the writer of the quoted paragraph never studied to any purpose.

This cartoon was published in The Boston Daily Globe (Boston, Massachusetts) of Sunday 26^th April 1891:

THE TAIL WAGS THE DOG.

President Harrison [cf. note 4] is at the head of his party, and Mr. Blaine [cf. note 5] is at the tail, but the recent proceedings at Cincinnati [cf. note 6], and other occurrences all over the country, indicate that this is one of the rare cases where the tail will wag the dog, and not the dog the tail.

Note 1: The term whisky ring denoted a combination of distillers and revenue officers formed to defraud the government of part of the tax on spirits. This was mentioned for example by The Brooklyn Daily Eagle (Brooklyn, New York) of Thursday 17^th March 1870:

Who are the whisky ring? The facts in outlines are, that the late Administration found its efforts to collect the taxes baffled. The government officers all professed to be very zealous, and yet somehow so much whisky escaped tax that the selling price in the market was usually lower than the single item of cost represented by the tax.

Note 2: On Saturday 5^th November 1870, The Brooklyn Daily Eagle (Brooklyn, New York) mentioned one of the raids conducted by the marines against illegal distilleries:

The Effect of the Whisky Raid on Webster’s Prospects.

The Sun this morning says:
Mr. Erastus D. Webster, who is running for Congress in the Third District against R. M. Whiting and Gen. Slocum, appears to have received his political deathblow by the suppression of the illicit whisky distilleries near the Navy Yard. The whisky men had been promised immunity in consideration of their support of Webster. They had been told that in case of Webster’s election a Supervisor would be appointed who would take especial care of their interests. Since the marines have smashed their stills and have put an end to their operations, the whisky ring do not take as much stock in Webster as they did; and it is believed that Mr. Webster does not take as much stock in the whisky ring as he did.

Note 3: Ulysses S. Grant (born Hiram Ulysses Grant – 1822-85), American general, was the 18^th president of the USA (1869-77).

Note 4: Benjamin Harrison (1833-1901), American Republican statesman, was the 23^rd president of the USA (1889-93).

Note 5: James Gillespie Blaine (1830-93) was an American Republican statesman. he refused the Republican nomination for President in 1888, supported Benjamin Harrison and later received an appointment as Secretary of State.

Note 6: During the meeting of the National League of Republican Clubs at Cincinnati on Tuesday 21^st April 1891, Joseph Benson Foraker (1846-1917), Governor of Ohio from 1886 to 1890, had praised James G. Blaine’s “magnificent administration”; this had been interpreted as injurious to President Benjamin Harrison, who was already unpopular among Republicans.