Did Rothschild say this famous quote? If yes, what did he mean by it?

What he (probably) meant by it (who can say for sure now?) is that of all the possible means a government has to control and influence its people, the issueing of money is the most powerful one. Even more powerful than law.

By creating new money, the government can decrease the value of the existing money in circulation, thereby lowering the buying power of those holding it. In turn it can use the created money to buy assets on the free market. It's not fair competition.

This quote is often used in (political) statements against fiat money. They reason that by inflating the money supply, governments are actually stealing the wealth from their people without them (directly) noticing it. It is much easier to just create new money than to raise taxes. Tax revolts were quite common back in the old days, but inflation revolts? None would be the wiser until it was too late. "Read my lips, I will not raise taxes.". He says nothing about national debt, eh?

The United States, before they became independent of England, had issues with their (forced) use of the English pound, a currency which they did not control. The US constitution actually spends quite some words on trying to prevent US money from ever becoming fiat. Money should belong to the people was the founding fathers' opinion, and not to the head of state. Thomas Jefferson has even been quoted as saying that

I believe that banking institutions are more dangerous to our liberties than standing armies.

So there you have it. Money is more powerful and dangerous than both laws and standing armies.

The saying is apocryphal and was originated by the populist author T. Cushing Daniel, a Washington-based lobbyist and lawyer, in his testimony before the U.S. Congress in 1911 in hearings on House Resolution 314 (whether financiers were restricting trade by domination of the money supply). This is what Daniel said:

William Pitt made this statement: "Let the American people go into their debt-funding schemes and banking systems, and from that hour their boasted independence will be a mere phantom." He realized the maxim that Rothschilds laid down as fundamental: "Let us control the money of a country and we care not who makes its laws."

It is true that the Rothschilds had "maxims" and these were published by London weeklies in the 1890s. All of these maxims were homey things like "Be sure you are right, then go forward." Cushing simply invented the imaginary "control the money" maxim for the purposes of his books and testimony.

The original Rothschilds were very religious, modest people. Its hard to imagine Nathan R. or his brothers as having said such an arrogant thing and it conflicts with what is known about his personality. Long after Nathan R. was dead, during the period 1890-1910, which was a period of anti-trust and anti-wealth agitation in the United States many writers made up all kinds of stories to portray the Rothschilds and other "barons" according to their stereotypes of rich, arrogant puppet masters. These stereotypes often did not resemble the actual people.

Concerning the phraseology and idea that Daniel expressed, this was not original to him, but, as you might imagine, was adapted from the works of others. In this case the phrase appears to have come from a short paper on the History of New York State published in 1892 by Welland Hendricks, a school principal. This is what Hendricks wrote:

Our Dutch forefathers who seemed to care little whether the flag of England or of Holland floated over the weak fort of New Amsterdam so long as their trade was uninterrupted have bequeathed their spirit to our keen-sighted non-voting business men of to-day, and all along the motto of our leading citizens has seemed to be — let us make the money of the nation and we care not who makes its laws.

Daniel completely perverted the original sense of the sentence to his own purpose. Hendricks was borrowing the phrase as well. In 1890, a meddlying clergyman named Wilbur Crafts, who tried to get all kinds of moralistic laws passed in New York, wrote a tract promoting the banning of work on Sunday containing the paragraph:

Massachusetts took upon herself the appointment of Boston's Police Commissioners, and so of her police. The lawless had been saying for years, "Let us appoint the city's police and we care not who makes its laws."

The actual origin of the printed phrase dates back to the 17th century Scottish Parliamentarian, Andrew Fletcher:

I said I knew a very wise man so much of Sir Christopher's sentiment, that he believed if a man were permitted to make all the ballads he need not care who should make the laws of a nation, and we find that most of the ancient legislators thought that they could not well reform the manners of any city without the help of a lyric, and sometimes of a dramatic poet.

-- An ACCOUNT of A CONVERSATION concerning A RIGHT REGULATION of GOVERNMENTS For the common Good of Mankind: In A LETTER to the Marquiss of Montrose , the Earls of Rothes, Roxburg and Haddington, From London the first of December, 1703'.

Now, of whom is Fletcher speaking? Who is the "very wise man"? It is, of course, Sir Phillip Sydney (1554-1586), the English poet who came to completely dominate the court of Queen Elizabeth even though he was only in his 20s. Sydney is the one who originated the phrase "Let me make the ballads of a nation, and I care not who makes its laws."