They are shown to be the only alien race humanity has made contact with and play a major role in the series from Forbidden Knowledge onwards.
They are analogous to the Borg in the Star Trek universe, both in their collective intelligence (or "mind union") and in their aim to conquer through assimilation (turning humans into members of their own species).
Vertigus's instructions were to "establish proximity with any alien vessel or base" and then broadcast the contents of a tape prepared specially for the occasion.
This seemingly-paranoid set of instructions was due to the unwillingness of Holt Fasner to reveal too much to any alien race "whose intentions were unguessable".
Vertigus however was willing to disobey these orders directly: something of an idealist, he resolved to travel to the Amnion ship and deliver the tape in person, mostly out of a desire to lay eyes on at least one Amnioni, but also in the hope of kick-starting diplomatic relations between the two races.
This is where the debate over 'first contact' started from: the Amnion's existence had already been known, and aside from matters of appearance, Captain Vertigus learnt nothing new about the race.
The Amnion had been discovered when the ship Far Star encountered a satellite of alien origin, containing mutagens, in a far extrasolar system.
Dissection reveals a fundamental genetic change - basic life processes remain intact, but protein and DNA structure are radically different.
Note: This list is not intended to be a definitive plot summary, more an overview of the parts the Amnion play, and it is assumed that the article on The Gap Cycle has already been read.
The Amnion repair the gap drive of Captain's Fancy, but sabotage it to test components that they, themselves, are developing that will allow their war ships to travel at near-light space-normal speeds.
They escape from Calm Horizons, Angus having sabotaged the super-light proton cannon by filling it with hull sealant, so it explodes when the ship attempts to fire.