The Booster Pack Rainbow refers to the cycle of background colors on Booster Pack sleeves followed by each series of OCG and TCG Booster Packs. This cycle was introduced in the OCG with Series 2, where it consisted of nine colors: green, yellow, red, orange, blue, purple, black, white, and brown. The brown was dropped after Series 3, and was not used in the TCG. The rainbow was introduced to the TCG beginning with Series 4, marking the beginning of the OCG and TCG receiving the same packs and thus following the same pattern.
So far, every Green set has had the word "Duelist" in its name since Series 4 (the Series 2 and 3 boosters instead had "Ruler" in their names), and has either introduced a new gameplay dynamic, or elevated a previous mechanic (Magic Ruler introduced Quick-Play Spell Cards, The New Ruler introduced the first archetype, Soul of the Duelist introduced "LV" monsters, Power of the Duelist introduced fast-paced Fusion strategies with "Future Fusion" and "Overload Fusion", The Duelist Genesis introduced Synchro Monsters, Duelist Revolution augmented Synchro Monsters by introducing Synchro Fusion Monsters like "Dragon Knight Draco-Equiste" and Return of the Duelist introduced an attribute-based series of boosters, being an EARTH-based attribute).
Likewise, a trend has been seen with the Black and White sets. Like their name, the Black sets introduce predominantly DARK monsters that alter the metagame at the time of that set's release (Shadow of Infinity released the Sacred Beasts, "Demise, King of Armageddon" and "Treeborn Frog"; Phantom Darkness released "Dark Armed Dragon"; Absolute Powerforce released "Battle Fader"; and Order of Chaos released the "Inzektor" archetype). Conversely, the White sets introduce predominantly LIGHT monsters designed to counter what was introduced in the prior Black set (Enemy of Justice released numerous monsters used in Counter Fairy Decks, Light of Destruction released "Honest" and the "Lightsworn" archetype, The Shining Darkness released "Herald of Perfection", and Galactic Overlord introduced the "Lightray" series).
The only other apparent/discovered patterns are that Brown packs feature a singular, powerful juggernaut-like monster, every Blue pack features a Dragon-Type (or Dragon-like) Cover Card (with the exception of Judgment of the Light), and Purple packs feature upgraded versions of past monsters with the exception of Photon Shockwave (Though its cover card shares many similarities with "Blue-Eyes White Dragon"). No trend has yet been noted for Orange and Red.