There are a vast amount of summons in this game, and there are many different ways to obtain them, with each having different uses. Here is my breakdown on them.
Common summons:
I call these common summons as these are the summons that most players will obtain in the game, they are:
Zigred
Pkhadd bear
Tiger
Iguraon
Atshi bat
Golem
All of those can be bought with city rep to be able to be used in battle, and called by opening your backpack when in a fight. There are also attack forms of zigred, pkhadd bear and tiger that can be bought from the arsenal with the right valour rank, these can be used outside of battle and cast at any player by inserting their name.
Rare summons:
These are what i call the summons that most players can still gain, but they are alot harder to get, these include:
The red summons from trade fair:
Kakurt
Large scorpion
King scorpion
Spider patriarch
Emperor scorpion.
Summons from chests:
Kretch
Grim gargoyle
Striagorn
These are all used in battle and summoned via backpack except kretch which are summoned by name insertion and can be done in or out of battle.
Quest/Event summons:
These are the summons that you can gain from doing various quests or regular events, these summons are mostly a once only obtain and include:
Valdagor the ripper
Phitzell (can be got once from event and once from quest)
Young yeti
Wild yeti
Scolopendra
The numbers of each of these obtained vary. They are all used in battle and summoned via backpack, some need a name inserted.
Card summons:
These are the summons that you can obtain from collecting a full set of cards, each set of cards will give different summon types and also some sets can have several summons possobile of which it selects 1 randomly. These are all summoned during fight via the backpack and include:
Elemental summon
Agudar wraith
Legendary warrior (2 of the card sets can give this)
Resurrected ghoul wraith
DefilerDO
All of these summons can be used once per week and reset at the start of each new week.
Rep summons:
This is the big one as there are many different reps and alot of them can allow various summons upon gaining enough rep.
City rep:
Gain 4000 city rep and you can buy an amulet to summon warriors, these require 3 mirrors to summon which can be bought or made from quicksilver. These can set marks on enemies which help the archers.
Gain 5500 city rep and you can buy an amulet to summon archers, these require 4 mirrors to summon.
*note, the common summons are also city rep based (except arsenal ones), but i made these seperate as they take much longer to gain and need lvl 8 and 9 respectively.
BoE rep:
Summons here include:
Cutthroat (500 rep)
Gordt ataman (1000 rep)
kretch butccher (2000 rep)
Chaos rep:
Summons here include:
Kia ling (500 rep)
via ling (1000 rep)
Dia ling (2000 rep)
Nia ling (medal of worship)
All of these summons may only be used once every 15mins.
Mercenaries rep:
Summons here include:
Mercenary Summon (medal of worship)
This can be done in either mercenary rep to aquire the summon.
Flaundin rep:
Sharkers (500 rep)
This summon may only be used underwater.
I think that about covers it all, but may have missed the odd summon here and there
All summons are called within fights unless i have said otherwise.