Become My Vassal of My Vassal
Peaceful vassalage is a somewhat different story. You can ask someone to become your vassal, although they will only accept if they're significantly smaller than you and with a significantly weaker military, and like you, and not always even then. You also can't make this request when at war. Civs that are willing to become your vassals might also make the offer themselves. I have found attaining peaceful vassalage extremely hard. If you are a builder, it's unlikely to succeed because you probably won't be big enough compared to the potential vassals, and the potential vassal also needs to be convinced you can protect them.
If they do agree to become your vassals, however, they will still be able to cancel this agreement. Just like regular deals, it can be cancelled after 10 turns if the vassal desires so - the master can't break vassalage deals, though. Capitulated vassals, on the other hand, can only break free if they lose half of their land since the time they became vassals, or if their population AND land is 50% of the master's. As you can probably guess, these are not very likely. If you make an AI capitulate, they will only ever hit the 50% thresholds if you get a good portion of your empire conquered.
When you gain a vassal peacefully, you also take the hit from increased maintenance. You also gain the ability to demand stuff, but if the civ agreed to be a vassal in the first place, it means you were good friends anyhow. And if so, you would have most likely been able to get some help from them by asking for it anyway.
I do not, admittedly, have enough experiences with vassalizing civs peacefully, having found that very hard, but from what experience I do have, it does not seem a particularly useful option. You have to be significantly larger and stronger than them, so you're not getting extra military protection – they are. You get increased maintenance costs and more danger of being drawn into a war you don't want to be in. Of course, you can also declare war yourself on someone and force the vassal to attack, too, though it's easier to declare without vassalization, and just bribe them to join the war if you need an ally.
Likewise, surrender isn't something that you should accept blindly. Carefully consider the fact that the civ's cities will then stay where they are, and they'll probably build more. If the civ has any Wonders or HolyCities, you forfeit any chance of ever owning them by accepting capitulation.
You'd think that, clearly enough, there has to be something else to gain from having vassals. And you're right – half the vassal's land and population count towards your Domination score. That is, if you control 40% of the world's land area, and vassalize a civ that controls 16%, your land area control will count as 48% for purposes of the domination victory. Thus, gaining vassals can put you closer to domination. Be careful, though – vassals can still build spaceship parts if allowed to advance that far, and win the game!
Vassalage often functions as a replacement for conquest. You don't have to dedicate the time and the troops needed to wipe out a civ completely, but you can settle for just making them capitulate. However, that is really more of a convenience thing or a shortcut. The really interesting part of vassalage is vassal deals among AI civs. That, along with a couple of other Warlords tweaks, makes some parts of the game more interesting.