When most of us think of "Do-Re-Mi," we think of the show tune from the 1959 Rodgers and Hammerstein musical The Sound of Music, which was made into a movie in 1965.  Within the story, it is used by Maria to teach the notes of the major musical scale to the Von Trapp children who learn to sing for the first time, even though their father disallowed frivolity after their mother's death.  Each pitch is assigned a syllable name in the song's lyrics. 

But Do-Re-Mi did not originate in the Sound of Music.  Its origins can be found in eleventh-century Italy.  The music theorist and Benedictine monk, Guido of Arezzo, invented a notational system that named the six notes of the hexachord after the first syllable of each line of the Latin hymn, “Ut queant laxis,” the "Hymn to St. John the Baptist," yielding ut, re, mi, fa, sol, la. Each successive line of this hymn begins on the next higher note, so each note's name was the syllable sung at that pitch of the hymn. 

Ut queant laxīs    resonāre fībrīs 

ra gestōrum    famulī tuōrum, 

Solve pollūtī    labiī reātum, 

Sancte Iōhannēs. 

"Ut" was changed in the 1600s in Italy to the open syllable Do, at the suggestion of the music theorist Giovanni Battista Doni, and Si (from the initials for "Sancte Iohannes") was added to complete the scale. In English-speaking countries, "si" was changed to "ti" by Sarah Glover in the nineteenth century so that every syllable might begin with a different letter.