Body Pieces: Spartan marked with (S) and Elite marked with (E).
(E) Ascetic Body - Unlock "Up Close and Personal" Achievement
(E) Ascetic Head - Unlock "Steppin' Razor" Achievement
(E) Ascetic Shoulders - Unlock "Overkill" Achievement
(E) Commando Body - Unlock "Triple Kill" Achievement
(E) Commando Head Unlock - "Overkill" Achievement
(E) Flight Body - Complete Tsavo Highway On Herioic or Legendary
(E) Flight Head - Complete Campaign mode on Heroic difficulty.
(E) Flight Shoulders - Complete The Ark on Heroic difficulty or higher.
(S) EOD Body - Complete Tsavo Highway on Legendary
(S) EOD Head - Complete Campaign mode on Legendary Difficulty
(S) EOD Shoulders - Complete The Ark on Legendary difficulty
(S) EVA Body - Complete Tsavo Highway on Normal or higher
(S) EVA Head - Complete Campaign mode on Normal
(S) EVA Shoulders - Complete The Ark on Normal difficulty or higher.
(S) Hayabusa Chest - Collect 5 hidden skulls
(S) Hayabusa Helmet - Collect 13 hidden skulls
(S) Hayabusa Shoulders - Collect 9 hidden skulls
(S) Katana - Complete 49 Achievements for at least 1000 gamerscore
(S) Mark V Head - Unlock "UNSC Spartan" Achievement
(S) ODST Head - Unlock "Spartan Graduate" Achievement
(S) Rogue Head - Unlock "Spartan Officer" Achievement
(S) Scout Body - Unlock "Too Close to the Sun" Achievement
(S) Scout Head - Unlock "Used Car Salesman" Achievement
(S) Scout Shoulders - Unlock "Mongoose Mowdown" Achievement
(S) Security Head - Earn 1000 Gamerscore points
Thanks to sythe343, Tetsuya_San, jaredt13, and pokelordguy for this information.