Maurian tells us:  "it's a well known fact that Paul was a chess genius when he was barely nine years old".

Morphy himself told Charles Stanley, at the American Chess Congress, that he had attended (at age 8) the Stanley-Rousseau chess match of 1845 (this match supposedly determined the first American chess champion) and that his Uncle Ernest acted as Rousseau's second.

Maurian gives us our earliest picture of Paul playing chess:

On a balmy summer afternoon Judge Morphy and his brother Ernest were seated on the back porch, which overlooked the long yard, playing chess. The game had bee a particularly interesting one, and lasted several hours, with the result that both armies were sadly reduced, though apparently still of equal strength. The Judge's king seemed in an impenetrable position and Mr. Morphy, after vainly checking and checking, wiped his perspiring brow and remarked that the game was a certain draw. Judge Morphy smilingly agreed with him and the pieces were swept aside to be reset for another trial. Now, little Paul, hardly out of skirts, had been an interested spectator to the closing stages of the drawn battle, and while the men were being replaced he astonished his elders by saying: "Uncle, you should have won that game."

Judge Morphy and Ernest Morphy looked at the boy and the former asked, "What do you know about it, Paul?" Paul, with the assurance of a born genius, asked leave to set the pieces in the final position, and, just to humor him, his father consented.  The boy faithfully and accurately arranged the men; and. then studying the board for only a moment, leaned forward and said: "Here it is: check with the Rook, now the King has to take it, and the rest is easy." And sure enough it was. The child had seen mate in an apparently impossible position, and the Judge and his brother simply stared at him, hardly able to express themselves in words.

 

In 1846, before the age of nine... The Evening Post relates this story:

Gen.Winfield Scott (famous hero of the Mexican War and first Commander-in-chief of the Union forces during the American Civil War) had many acquaintances there (at a chess club on Royal St. ^{see note}), some of them quite intimate, and knowing the habits of the members he repaired to their very comfortable rooms within a few hours after reaching the city. It may be said to have been one of his vanities as well. He was in the front rank of amateurs in his day....he turn to Chief Justice Eustis and asked whether he could play a game of chess in the evening...."I want to be put to my mettle!" "Very well," said Justice Eustis, "We'll arrange it. At eight o'clock tonight, if that will suit you."
At eight o'clock, dinner having been disposed of, the room was full. Gen. Scott, a towering giant, was asked to meet his competitor, a small boy of about 10 (actually, he was eight and a half) and not by any means a prepossessing boy, dressed in velvet knickerbockers, with a lace shirt and a big spreading collar of the same material.
At first Gen. Scott imagined it was a sorry jest, and his tremendous dignity arose in protest. It seemed to him that his friends had committed an incredible and unpardonable impertinence. Then Justice Eustis assured him that his wish had been scrupulously consulted; that this boy was....quite worthy of his notice, So the game began with Gen. Scott still angry and by no means satisfied. Paul won the move and advanced his Queen's rook's pawn. In response to the General's play he advanced other pawns, Next he had two knights on the field; then another pawn opened the line for the Queen, and at the tenth move he had the General checkmated before he had even begun to develop his defense. There was only one more game. Paul Morphy, after the sixth move, marked the spot and announced the movement for the debacle - which occurred according to schedule - and the General arose trembling with amazement and indignation. Paul was taken home, silent as usual, and the incident reached the end.
The few survivors of that era still talk of Paul Morphy's first appearance in public, but only by hearsay. Gen. Scott lived to wonder that should have ever played with the first chess genius of the century, or for that matter, of any other century.

 

From the New Orleans Times-Democrat :

...Well do we remember seeing him (Morphy) from the street playing chess with his grandfather, Mr. Le Carpentier, in the latter's counting room, situated in the lower story of his residence. The boy was small, and the ledgers or other of grandpa's commercial books had to be piled under him to enable him to sit at the required height to the table; and when we thus saw him we did not know, but learned afterwards, that the grandson was all the time giving grandpa the odds of a rook and beating him like Old Harry.

 

In 1850, Paul played the Hungarian master, Johann Jacob Löwenthal and, while there has been some confusion about the results, it's generally believed that Morphy won all three games. Some believe one game was drawn. From Bretano's Chess Monthly:

Paul, he says, was a little fellow and stood up to the table. Mr. Morphy and his brother, Judge Morphy, the father of Paul, and Rousseau, were lookers-on. Lowenthal was one of the most noted and scientific players in the world, and a finished, courteous gentleman. He at first supposed that the game would be a bagatelle, but Mr. Morphy told me that as he, Lowenthal, got into the game and felt Paul's force, his startled look and upraised brows after each move of Paul's was perfectly ludicrous or as Mr. Morphy in his French vernacular expressed it, comique.

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Paul Morphy's School Days