Who is Alphinaud and who is Gaius?

[Cilia]

Alphinaud is the grandson of Louisoix, a Sharlayan sage who came to Eorzea in order to try and forestall the Seventh Umbral Calamity that played out during Legacy. During 2.x he is largely leveraging his grandfather's influence to get his way. He's also an academic prodigy from one of the most prestigious schools in the known world, along with his sister Alisae.

He's rather vainglorious during that time period, but the Crystal Braves betray the Scions during 2.55, and the experience is humbling enough that he has largely left said vainglory at the door since.

You do get to meet their father Fourchenault much later in 5.55, and their mother is mentioned, but not seen.

Gaius hails from the Garlean Empire, a fascist... empire from another continent to the far north. Their race, the Garleans, are incapable of using magic and were pushed to the inhospitable territory they now call home, so they eventually developed magitek to even the scales and feel they have an axe to grind with everyone else because of it. Their emperor has outlawed religious observation of any kind due to the dangers posed by primals, further decreeing they must conquer the world to completely stamp out the threat posed by primals, and they follow a very "might makes right" philosophy due to their harsh homeland and the circumstances that led to their militarization.

Of course, very little of that is so simple - the first Garlean Emperor was actually an Ascian in disguise, who quickly revolutionized Garlemald into the warmongering and oppressive empire it is today by introducing them to magitek and spinning various lies.

In regards to Gaius, he is one of the better Garlean legates - he follows the meritocratic values preached by the Empire unlike many of his contemporaries. He survived his defeat in 2.0, reappears in 4.4 / 4.5, and is the main protagonist of 5.x's "Sorrow of Werlyt" trial storyline, having learned both the error of his ways and the lies spun by the first Emperor. He now seeks to make amends and save his adopted children from one of the most unhinged legates thus far.

Addendum: the XIVth Imperial Legion was defeated. Garlemald has 13 more waiting in the wings; besides which the XIVth went rogue during the events of A Realm Reborn. You will meet more legates, and the Emperor himself, in due time.