Consider a scenario where I land 2 Alien Ships at a Location with a Resistance level of 1. This means there will be 1 Fight.
Another very important set of terminology to differentiate are the terms Battle, Fight, and Fight Round. A Battle is all of the Fights at a given Location, a Fight is single combat against one Human Resistance (or a grouping of enemy Aliens), and a Fight Round is one round of combat between the Aliens and the Resistance to see if either side is defeated. So one Fight is a subset of an entire Battle, and a Fight Round is a subset of a single Fight.
As an example, if your Aliens attack a City Location with a Resistance Value of 3, which would be a Battle involving 3 separate Fights (one Fight for each of the Resistance). Each of those Fights would then be resolved in a series of Fight Rounds.
At the conclusion of that Fight Round, the Resistance has a higher Strength Total. That means that they win the Fight Round and I lose an Alien ship.
Whichever side has the higher Strength Total (including Fight Dice rolls) wins the Fight Round.
Since that is the only (and, therefore, final) Fight of the Battle, the fighting should be completed for my turn, I believe, because it says may, below. This would only be true if there were more Fight Rounds to be completed.
From the Resistance Victory section of the Rules WIki (Emphasis added by me):
If the Resistance wins the Fight Round, 1 of the Alien Saucers involved is Destroyed and returned to the player’s Alien Pool. If there are still Aliens at the Location, you may start a new Fight Round. If there are no Alien Saucers left at the Location, the Battle is over and the Aliens have lost (any remaining Resistance cards are discarded). Next time this Location is attacked by Aliens, a brand new Battle must be fought from the beginning; with entirely new Resistance (any Fights previously won here are ignored).
Does this simply mean that the remaining Alien Ship stays on that Location (and can therefore start the attack over again on my next turn), or does it mean that I have survived all of the Resistance that the humans can throw at me and I have therefore conquered that Location (and should place a Conquest Marker on it, gaining the associated Terror Points)?
P.S.: If someone with 150+ Reputation could add "conquest-of-planet-earth" as a tag, I would greatly appreciate it. Thanks!
EDIT: thesunneversets response got me thinking...
The Exploring Empty Spaces section says...
to conquer a Location you must fight a number of Human Resistance equal to the Resistance Value of that Location."
...but it doesn't say you must win the fight, just that you must fight them. Later on in the Battle at an Existing Location section, it says...
If you have one or more Aliens at a Location with a Resistance Value of 1 or higher that has not been conquered (no Conquest Marker on it), you must fight a Battle there as described in the section on Fights below.
...and similarly in the Fight any Enemies at a Location section,
If you have one or more Aliens at a Location (or Landing Site) with one or more Resistance Counters or enemy Aliens, you must resolve a Battle at that Location as well.
Because these rules seem to indicate that an Alien Ship could be on a Location without explicitly having moved there in the same turn, I'm inclined at this time to think the Alien Ship can stay there, but the Conquest Token doesn't get placed.