In the diplomacy menu, things on the left side are things that you are offering. things on the right side are things the enemy is offering.

"Offer surrender" means you are placing an offer of your surrender on the table. You are the one giving up.

For the enemy to surrender, there's a "Demand surrender" option. which makes them surrender if they accept it. they very very very rarely do this.

i once managed to make magnar surrender, after taking all but one of his cities, and having an army on the doorstep of the last one. 

The result of this is that Magnar became a champion of mine - a unit i can control and move around like any other. Although he also gains a new trait (broken soverign or something like that) which gives him significant penalties to many stats. As far as i can tell, every other champion, city, outpost, and resource he had, was simply destroyed, not transferred over to me.

Getting another sovereign to surrender is usually a great coup and very worthwhile.  First he doesn't cost any gold to recruit and he doesn't even cost 1 gold each turn in maintenance.  Then he usually comes with a load of artifacts and other goodies he's picked up from questing. Probably the best haul of loot you'll ever get The biggest benefit is that his faction prestige is added to yours, so all your cities and his old ones benefit in terms of growth.

In my current game with Procipinee I have gained Oracle Ceresa from a Demand Surrender and he's proved quite useful.  Access to Death spells is useful for a Life mage (Blind, Curse and Wither) and his spell Horrific wail is deadly as it depends on level.  The penalty he has called Broken Spirit is -25% Accuracy, -25% Spell mastery and -25% HP.  The Spell Mastery hit is painful as it makes most spells more resistable, but I am levelling him up and using Destiny's gift to boost a few stats.  He's not my best support champion by an means but he's certainly not rubbish.

JJ

Yes, well, enemy sovereigns seem to give up and surrender when your power rating is about 10x times theirs, effectively meaning you have destroyed most of his/her troops, conquered most cities and that their power rating is basically very low ( = no units, of course).

Getting vassals has significant bonuses, as mentioned above:

+ You get their traits, like that armorer. Or in Relias' case, you get to recruit champions for free. And Karavox gives you the Silver Tongue ability. You also get the negative traits, like Cruel.

+ On harder difficulties, enemy sovereigns have extra traits, like Merchant, Strenght, Mighty etc. They can be quite powerful with attack power on their own.

+ They have completed quests based on calculations. I have often had them carry stuff like Arcane Armor, Scythe of the Void and that mauling broadsword.

+ Prestige is a great add, as mentioned.

So far, I've been able to subdue everyone but Ceresa in my various games. For some reason, she is one tough nut to break and usually just ends up ramming against a monster after having no cities, and just dying. I usually play on hard difficulty, and at least on that difficulty setting they have fine bonus traits.

I would expect them to give up their cities and outposts too, rather than destroying them in the process. I'd like to see them surrender to other AIs too, once in a while, and give their assets to them. But you can't have everything I guess.

I turned "Surrender frequency" (or what that option during game setup is called) to "often" and I got Ceresa to surrender with 1 city left.

The city was destroyed in the process, which seems very counterintuitive to me. I'd say it's a bug.

The default requirements for AI surrender are fairly preposterous- many times I have reduced an AI to a single city and almost no military, yet they refuse to surrender (presumably because my own force, while deadly, is fairly small as well).  I'd prefer it if there were two surrender mechanics- vassalization, where the AI retains its cities but essentially joins you in a permanent alliance where it has autonomy over its own cities but the follows the player's lead in foreign relations and research, and complete capitulation, which would work like the present system.  The former would have a significantly lower threshold to achieve than the later, but perhaps have some downside, such as the potential for your vassal to backstab you if your empire becomes too vulnerable, or increased costs for the peaceful win conditions.