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", Return of the Duelist introduced an attribute-based series of boosters, (this one being an EARTH-based booster pack), and Duelist Alliance introduced Pendulum Monsters.
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 Phantom Darkness released "Dark Armed Dragon"; Absolute Powerforce released "Battle Fader"; Order of Chaos released the "Inzektor" archetype; and Legacy of the Valiant released "Gravekeeper's" support. 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", Galactic Overlord introduced the "Lightray" series, and Primal Origin introduced the "Artifact" archetype.).
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) that was owned in a major character in the anime, 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"). Orange packs feature the ace cards of the non-main characters (with the exception of Extreme Victory) and every Cover Card on the Yellow packs after Rise of Destiny feature the ace monsters of several main characters of the series, of which all of them are either upgraded versions or counterparts of monsters that each respective character also owns and uses. No trend has yet been noted for Red packs.