From Halopedia, the Halo wiki
The Halo Theme is the staple music of the Halo series.
It is characterized by its opening Gregorian chant, a tribal percussion section, and a low strings section. The low strings section featured a series of triplet rhythms, in which each triplet was followed by the melody leaping upwards by an octave, a ninth, a tenth, and finally an eleventh, before restarting at an octave. The complete theme features high strings playing a high-tempo melody which repeats in variations.
Martin O'Donnell and Michael Salvatori composed the Halo Theme over the course of three days in July of 1999. O'Donnell was asked by Joseph Staten on a Thursday to compose a soundtrack for the then-upcoming Halo MacWorld premiere demo on the following Tuesday. However, the music would need to be first recorded and shown internally within Bungie on Monday — the same day Bungie representatives would board the flight to New York for the presentation on the following day. 
Staten told O'Donnell that the theme needed to evoke the game's "ancient, epic and mysterious" atmosphere. To best convey this impression, O'Donnell drew inspiration from his studies on Middle Ages music and decided to incorporate Gregorian chant into the theme, which has since become a staple of the Halo series in its various forms. He was also inspired by the first verse of The Beatles' song "Yesterday", which inspired the four-phrase structure of the Gregorian chant. O'Donnell has stated that he laid down the core melody of the theme in about half an hour during the drive to the studio at Salvatori’s house.
Recording one day prior to the MacWorld presentation, the theme was recorded with an orchestra composed of six string instruments: four violins and two cellos. O'Donnell and Salvatori sung the Gregorian chant with three jingle singers, who had previously collaborated with the duo on a number of commercials. O'Donnell and Salvatori applied reverb to the strings and choir recordings, while the cello recordings were overdubbed, thus creating the impression of a large orchestra when the recordings were added on top of the MIDI recording. O'Donnell requested one of the jingle players to sing the Qawwali-like solo during the string melody, but upon hearing O'Donnell's example, it was decided O'Donnell would sing the solo instead.
After the MacWorld presentation, O'Donnell adapted different aspects of Halo Theme into various pieces of music in Halo: Combat Evolved and its sequels. O’Donnell cited an anecdote from his teacher, film composer David Raksin: according to Raksin, when he expressed concerns over the repeated use of the main theme of Laura in its film score, Raksin was told by his mentor Alfred Newman that good themes should be used repeatedly, while bad themes would have never been used in the first place. O’Donnell and Salvatori followed this philosophy when composing other Halo music - a tradition carried by future composers in the franchise. Called “emotional equity” by O’Donnell, a sense of familiarity would be felt by players when themes are being repurposed, remixed, and reused throughout the trilogy.
Halo: Combat Evolved
In its original incarnation, first heard in the Halo announcement trailer, it placed heavy emphasis on deep, powerful drums and fast paced strings, mainly the violin.
The track plays:
The Gregorian chant was adapted and incorporated into the following tracks:
The string melody was adapted and incorporated into the following track:
The original piece from Halo: Original Soundtrack was remixed into the MJOLNIR Mix, the first track on the Halo 2: Original Soundtrack, Volume 1. The main theme remains unchanged, but it features electric guitar overlays by former Whitesnake guitarist Steve Vai. Halo 2 uses the track in its original form in certain parts, such as at the beginning section of the level Metropolis. The MJOLNIR mix is played at the end of Metropolis as John-117 boards the Scarab, and later during the end credits.
Halo 2 featured Cloistered Expectancy (from the track Prologue), which was the Halo 2 rendition of the Gregorian chant from the Halo Theme.
Two variants of the Halo Theme are featured in the Halo 2 Anniversary Original Soundtrack. The Halo Theme Gungnir Mix is a remixed version of the MJOLNIR Mix from Halo 2, and the Halo Theme Scorpion Mix is a percussion heavy version of the main theme.
The theme was again revised for Halo 3, arranged into the track One Final Effort, this time recorded with a live orchestra instead of synthesized strings with the grand piano as its main instrument. The version used in Halo 3 finishes with the ending used at the end of The Maw, as opposed to the reprise of the opening chant used in the original theme and the MJOLNIR Mix. Occasionally, 343 Guilty Spark can be heard humming it with the IWHBYD skull activated in campaign.
The Halo Theme was reprised as the song Sacrifice. The chant portion of the theme, now referred in the Halo 4 credits as "Halo Cantorum", is heard when Installation 03 is revealed in the level Composer. The main notes of the Halo Theme play in the theme 117 when John neutralizes the external shield generators on Mantle's Approach.
Notes that ambiguously sound similar to the main chorus play subtly when Dr. Halsey tells her interrogator not to underestimate the Spartans.
Halo 5: Guardians
The Halo Theme returns in Halo 5: Guardians, rearranged as the The Trials with more focus on electronic digital instrumentation. It's also featured as part of the soundtrack in Halo Canticles. It was recorded at Abbey Road
The Halo theme was featured in the Discover Hope trailer, implying it will have its own rendition in the sixth mainline Halo game.
Part of the theme plays during the song Set a Fire in Your Heart.
The Halo Theme does not appear in Halo 3: ODST. While stated before release that the Theme would not appear in Halo: Reach, several tracks in the soundtrack borrow motifs from it. For example, Immemorial from Overture takes three bars from the beginning of the chant, while Unreconciled from Tip of the Spear takes the drumbeat and uses both the beginning of the chant and the main melody.
A partial remix of the track appears in the Halo Legends Original Soundtrack. This remix plays during Halo Legends episode Origins during the scenes depicting the escalation of the Human-Covenant War.