After you've captured enough of his planets and/or after enough time has passed, he will send you a message saying he's tired of war.  I don't know of any way for you to prompt that, or to ask for peace, except perhaps by proposing to end all wars in the United Planets.

That might very well be, I've never had it pass when I've been at war.

"You must be joking" is the response when they need something in return; try offering something - credits, preferably, but they'll take technology, ships, starbases, resources, whole colonies, whatever.  Also, consider bribing their neighbors to declare war on them; your enemy might divert forces away from you to fight their new enemies for the next fifty turns, and may make them more amenable to making peace.

When I play, I have tech trading turned on but players can only trade tech they've discovered, so I make a point of giving away farming and economic technologies to minor races which will increase their raw production - then I bribe them to make war on my enemies, and it's cheap.  Actually, I play the role of Galactic Loki - if I'm not fighting another race I bribe them to fight their neighbors.  It permanently damages their relations with each other (so they're less likely to ally against me), it keeps them busy, and saps their resources.

I managed peace with another Faction by simply giving it almost everything I had.

You can make peace with a faction the same way you would trade with them. However, if you have a lower military rating, they will demand much, more more from you to accept the trade. You are basically trading for a peace treaty. "You must be joking," means the total value of the items that you have offered (regardless of whether planets, ships, tech, or whatever) is extremely insufficient. The greater the difference between your rating, the more you will need to offer. If the difference is too large, you can offer everything you have and still not be enough. Often times, it is simply not worth it to ask for peace. Just destroy them if you can.

Naturally, the other way is to blast them into oblivion. Peace.

Also, I have found that it is a waste of resources to bribe factions into fighting with each other because as soon as they can, they will usually sign a peace treaty.

I did bribe two other factions into declaring war on my enemy.  They then just ignored him while he kept pounding on me.

I never have a problem keeping the peace in the early game. I just make sure I have more and better planets and I do a lot of trading, granting open borders and diplomatic stuff.

I seldom play over 400 turns and by t300 somebody is going to DW. I am usually quite a bit stronger militarily and they sue for peace in 4-5 turns, rinse and repeat.

Currently the game is not designed for my preferences. It is fine for a conquest game where you build your military asap and go after the AI with a strategy to take them all out in the shortest period of time.

My preference is to win by influence, fighting a few wars along the way just for the fun of it. I like to play on the big maps and it just takes too long to win by influence

This is not entirely true. You can just buy out their best planets relatively close to the center of their ZoC, and then turn those planets into culture planets.

All planets besides their home world are fair game, but they can get expensive. Nevertheless, by turn 250-300, I am usually making enough money to buy up to three class 20+ planets per trading window on pure cash.

That always kind of annoys me. It just seems wholly unrealistic that you`d sell 3 decent worlds of your nation for money. It shouldn`t be just about money, but the fact they`re selling a whole PEOPLE`S, even Races of your people. Could you imagine anyone of a Country today just selling off part of their populated country? England sell manchester? The US sell Wyoming? There should also be strategic factors to selling a planet as well- And how happy the people are. Even if the people aren`t involved, it`s hugely difficult to relocate people who don`t want to move from a village, let alone a planet?

It should be really hard to buy a planet. I think they should change the calculation so a Civ would be very, very reluctant to sell a world unless it was pretty much useless and cost a good 2\3rds or more of the Buyer`s income, no matter how rich the Buyer gets.

That said, I`m ok with the people of a planet `changing-sides` because they become influenced by a better-off neighbour.