The words are said to have been written in 1847 by a marine stationed in Mexico during the Mexican-American war. (They were slightly changed in WW II when aircraft were added to the Marine Corps)
The tune is from a comic opera 1st performed in 1868, written by Jacques Offenbach. Offenbach is also known for writing the "Can-Can". The tune and words were first put together in 1919. The Marine Corps copyrighted it in 1920.

The song was originally written by field artillery First Lieutenant [later Brigadier General] Edmund L. "Snitz" Gruber, while stationed in the Philippines in 1908. Interstingly, Gruber was related to Franz Gruber, who composed "Silent Night,".
The original title is the "Caisson Song." The original lyrics reflect routine activities in a horse-drawn field artillery battery. The song got its beginnings during a difficult march across the mountains. As a lieutenant leading a small detachmen, Gruber overheard a section chief call to his drivers, "Come on! Keep them rolling!"
In March 1908, about a year after Gruber overheard that section leader, six young lieutenants - including William Bryden and Robert Danford - gathered in his thatch hut and decided they needed a song for the field artillery. "A guitar was produced and tuned and - in what seemed to us a few moments - as if suddenly inspired, Snitz fingered the melody of the now-famous song," recalled Danford. Danford and Bryden helped complete the lyrics. Gruber taught the song to officers as they arrived in the Phillipines. According to Danford, its popularity was instantaneous, and almost in no time all six of the regiments then composing the U.S. Field Artillery adopted it.
During World War I, senior artillery leaders wanted an official marching song. An artillery officer who did not know Gruber and thought "The Caisson Song" dated back to the Civil War, gave the piece to noted composer and bandmaster John Philip Sousa and asked him to fix it up.
Sousa incorporated Gruber's piece into his composition, which he titled, "The U.S. Field Artillery March" - a few beginning measures being his own and the balance from Gruber. The resulting song became a blockbuster record during World War I, selling about 750,000 copies. Gruber heard of it and asked Sousa, "How about some money, since I wrote the song?" Embarrassed, the innocent Sousa made certain Gruber got his royalties.
In 1948, the Army conducted a nationwide contest to come up with its own official song. None of the five winners achieved any notable popularity. In 1952, the Secretary of the Army appealed to the music industry for a composition. Composers submitted more than 800 songs, but none were accepted.
So a soldier music adviser was asked to try his hand at it. As a result, H.W. Arberg adapted "The Caisson Song" to become the official U.S. Army song, "The Army Goes Rolling Along." The song is played at the conclusion of every U.S. Army ceremony and all soldiers are expected to stand and sing.

The original words and music were written by Captain Francis S. Van Boskerck, USCG in 1927. The current verse, and a second chorus, were written by Homer Smith, 3rd Naval District Coast Guard quartet, Chief Cole, others and LT Walton Butterfield USCGR in 1943, during WW II. The section most known today and used in MOTAF is actually a "chorus"

1st chorus (original, 1927 version)

The word "weigh" in this sense comes from the archaic word meaning to heave, hoist or raise. "Aweigh" means that that action has been completed. The anchor is aweigh when it is pulled from the bottom. This event is duly noted in the ship's log.
Lieut. Charles A. Zimmermann, USN, a graduate of the Peabody Conservatory of Music in Baltimore, had been selected as the bandmaster of the Naval Academy Band in 1887 at the age of 26. (His father, Charles Z. Zimmermann, had played in the band during the Civil War years.)
He was popular with the midshipmen and known for composing a new march each year for each graduating class. In 1906, Lieut. Zimmerman was approached by Midshipman First Class Alfred Hart Miles with a request for a new march, one with a swing to it so it could be used as a football "fight" song.
Supposedly, in November 1906, Zimmerman wrote the music and Miles wrote the 1st 2 verses while the 2 men were seated at the organ in the naval academy chapel. This march was played by the band and sung by the students at the 1906 Army-Navy football game later that month, and for the first time in several seasons, Navy won.
This march, Anchors Aweigh, was subsequently dedicated to the Academy Class of 1907 and adopted as the official song of the U.S. Navy. The words most often sung today are actually from the 2nd verse of a revised set of words by George D. Lottman

In 1938, Liberty magazine sponsored a contest for a composition to become the official Army Air Corps song. Of 757 scores submitted, Robert Crawford's was selected by a committee of Air Force wives. The song (informally known as "The Air Force Song" but now formally titled "The U.S. Air Force") was officially introduced at the Cleveland Air Races on September 2, 1939. Fittingly, Crawford sang in its first public performance.