What are the major factions and players on the Magic and the Science side? There are various factions that exist on both the Magic and Science side, but what are the most prominent ones and who are their most well-known members?
Here is a generalized list of the people and faction in both the anime and manga continuity (including a bit from the light novels).

Magic Side

Roman Catholic Church


Matthai Reese, Pope
Biagio Busoni, Bishop
Ridovia Lorenzetti, Mardi Gras


# God's Right Seat


Fiamma of the Right
Acqua of the Back/William Orwell, Flere210 (Saint)
Vento of the Front
Terra of the Left


Russian Orthodox Church


Nikolai Tolstoy, Archbishop


# Annihilatus


Vasilisa, Head of Annihilatus
Sasha Kreutzev


Anglican Church


Laura Stewart, Archbishop
Orsola Aquinas 


# The Church of Necessary Evil (Necessarius)


Kaori Kanzaki, Salvare000 (Saint)
Stiyl Magnus, Fortis931
Index Librorum Prohibitorum, Dedicatus545
Tsuchimikado Motoharu, Fallere825
Sherry Cromwell, Intimus115


# (Roman Catholic, London Branch) Agnese Forces


Agnese Sanctis, Leader
Angelene
Lucia


# Amakusa-Style Remix of Church


Kaori Kanzaki, Pontiff/Priestess
Tatemiya Saiji, Substitute Supreme Pontiff
Itsuwa
Uragami


Knights of England


"Knight Leader", Leader


British Royal Family


Elizard, Queen of England
Riméa, Princess of England (Oldest)
Carissa, Princess of England (Second-Oldest)
Villian, Princess of England (Youngest)
Silvia, Head Maid of the Royal Family (Saint)


Mercenarys


Oriana Thomson, Basis104




Science Side

Board of Directors


Aleister Crowley, General Superintendent
Kaizumi Tsugutoshi, Chairman
Oyafune Monaka, Chairwoman
Shiokishi, Chairman
Thomas Platinaburg, Chairman
Kumokawa Seria, Advisor


Hound Dog


Kihara Amata, Commander


Misaka Network


Misaka Imouto, Subject No. 10032
Last Order, Subject No. 20001


A Certain Magical High School 


Kamijou Touma, Imagine Breaker (Level 0)
Himegami Aisa, Deep Blood (Gemstone)
Tsuchimikado Motoharu, Auto-Rebirth (Level 0)
Aogami Pierce
Tsukuyomi Komoe, Chemistry Teacher
Oyafune Suama, Math Teacher 
Yomikawa Aiho, P.E. Teacher


Tokiwadai Middle School


Misaka Mikoto, Electromaster (Level 5 Esper, No. 3)
Shirai Kuroko, Teleporter (Level 4)
Shokuhou Misaki, The Queen of Tokiwadai, Mental Out (Level 5 Esper, No. 5)
Kongou Mitsuko, Aero Hand (Level 4)
Wannai Kinuho, Hydro Hand (Level 3)
Awatsuki Maaya, Float Dial (Level 3)


Sakugawa Middle School


Uiharu Kazari, Thermal Hand (Level 1)
Saten Ruiko, Aero Hand (Level 0)


SCHOOL


Kakine Teitoku, Dark Matter (Level 5 Esper, No. 2) (Awakened)
"Girl in the Dress", Measure Heart (Level ?)
"Boy with Goggles" 


GROUP


Accelerator, Vector Change (Level 5 Esper, No. 1) (Awakened)
Tsuchimikado Motoharu
Musujime Awaki, Move Point (Level 4)
Etzali


ITEM


Mugino Shizuri, Meltdowner (Level 5 Esper, No. 4)
Takitsubo Rikou, AIM Stalker (Level 4)
Kinuhata Saiai, Offense Armor (Level 4)
Hamazura Shiage, Grunt, Chaffeur, and Informant


MEMBER


"Professor", Leader
Baba Yoshio
Xochitl
Saraku, Kill Point (Level ?)


BLOCK


Saku Tatsuhiko, Leader
Teshio Megumi


Judgement


Konori Mii, 177th Branch Office Chief, Clairvoyance (Level 3)
Shirai Kuroko
Uiharu Kazari


Skill Out


Hattori Hanzou, Leader (Temporary) 


# Big Spider


Kurozuma Wataru, Founder, Former Leader
Hebitani Tsuguo, Second Leader


Anti-Skill


Yomikawa Aiho, Lieutenant, Branch 73
Tessou Tsuzuri
Saigou Ryouta


Kihara Family


Kihara Gensei
Therestina Kihara Lifeline, (Former) Captain of MAR
Kihara Amata, Leader of Hound Dog 


Imaginary Number District


Kazakiri Hyouka, Counter Stop 


Unaffiliated


Sogiita Gunha, Attack Crash (Level 5 Esper, No. 7) (Gemstone)




Other Side


Gabriel, Archangel
Aiwass, DRAGON