Espers

Espers, or Eidelons, are no gods - but demi-gods at best, meaning that they are not immortal. Powerful creatures that can be summoned through a series of means. But whatever means these take, the actual Espers are never actually removed from the Feymarch, their home. Most of the Espers which were once forced into Magicite and were 'killed' according to one world, still existed in the other, and so they were reformed into the Feymarch with the formation of the World of Ruin. That is to say, those Espers that were killed and turned into Magicite in the World of Magi (FF6), are back in Feymarch again and alive - but their Magicite still remains behind.

Standard Summoning

Someone who can call or summon an Esper, truly only calls forth a mote of their power. A part of the Esper is projected into their world from the Feymarch, and after this action is done, that part comes back to the Esper. They have no recollection of the summoning in most cases - though there are always rare exceptions. In the case of casting an Esper forth from a Jewel or Magicite, it simply serves as the 'focus' for this power.

Dark Summons

One can summon dark variants if one has a dark heart. This does not afflict the original Esper with a darker soul - and is just the same as a Standard Summoning for most intents and purposes.

Aeons

Aeons are dreams of Espers from another world. Though Zanarkand lies in ruins in this world, the Aeons continue to dream up their version of the place. And with that, there remains to be the ability to summon these memories forth. An important thing to note however, is that with the connection of the portals and some of Spira's World Shards having landed onto different worlds, summoners may have a heck of a trek on their hands to get the Aeons they desire... and may found the dreams of these beings rife with some rare new creations. (Creating minor Aeons is allowed, based on Monters and other Espers - if you have other ideas, talk to a staffer.)

Guardian Forces

The Espers do sometimes travel outside of their world, and that is to form the rare contract to become someone's Guardian Force. However, beyond this personal interaction with the 'summoner' or 'caller', there is no difference from standard summoning. However, the 'part' that the Espers give up to these people in Junctioning with them does grant a certain sense of 'relationship' between the two.