There are four Dragon Age Origins romances that you can pursue. Note that this does not serve a noticeable material difference for the game. You'll just get some new dialogue for your ending and you'll get to see one of the character's in their underwear. It also comes in a bit handy since forming a relationship is an easy way to get a lot of approval quickly. The four characters with whom you can have a relationship are: Alistair, Zevran, Leliana and Morrigan. Leliana and Zevran swig both ways while Morrigan is just for male characters and Alistair is just for females.
You can pursue a relationship with different people at the same time. They will eventually confront you and make you pick. This naturally costs you some approval with them, but it isn't that bad.
Also, before we start, I'll note that the expansion pack, Dragon Age: Awakening, features no romance options. If you have an existing romance from the first game, then it will be mentioned in the epilogue.
Winning Leliana's heart is actually pretty easy. I had her at 100 about halfway through the game. Just make sure that you don't side with the cultists during the "Urn of Sacred Ashes" quest. She doesn't like that and will attack you if you defile Andraste's ashes. If she's not there, then you should be able to lie to her and convince her otherwise. Even if you "harden" her (see below), she will need a special conversation path to talk her out of defending Andraste.
To start the romance with Leliana, you need to flirt a bit. Female wardens can compliment her hair. Male wardens can make some horrible pickup lines after asking why she was in the chantry. You have to love lines about "forbidden fruit." If you mess things up somehow, you'll get another chance after you finish her companion quest. You just have to say that you want to talk.
Gaining approval with Leliana is very easy. You can just talk to her and be nice for the most part. Agree to listen to her stories and compliment her storytelling abilities. Even if you don't like the chantry or believe in the Maker, she does. She will not appreciate you insulting either or questioning her faith. You can also give her gifts and most of her gifts are bought or found easily. I won't list them all since we have a wonderful Dragon Age gift guide already. If you do ever run into the traveling dwarf merchant, he sells a pair of nice shoes for very cheap. This are a very good gift for Leliana and you're bound to run into the merchant an annoying amount of times during the course of the game. The best gift is to give her a nice pet from Orzammar. If you're already fairly friendly with her, then she should mention that she thinks the little nugs in Orzammar are cut. This will open up the quest "A Cute Nug". She will be very grateful and you'll get a big boost to approval for basically nothing.
Once she's past 71 points she's willing to give you kiss at the camp and after she's fully in love with you she'll be willing to go all the way. That's about it for her. I will note that this is a good romance for practical reasons, since high approval will help turn Leliana into a crazy bard assassin.
Romance in Dragon Age Origins – Zevran
Zevran is also a fairly easy relationship. He actually flirts with you first, so all you have to do is not shoot him down and then return the compliments. Once again, you just have to be nice to Zevran. He doesn't actually respond to too many of the quests, she you should be fairly safe dragging him around with you if you want. He really likes gold and silver bars and a nice leather gifts. You can find a pair of Dalish Gloves in the Brecilian Forest. Some Antivan Boots are in the shop in Haven. They're in a chest by the dead knight.
Other than that, just talk to him about his past missions and compliment him. After gaining some approval he will be willing to hook up with the warden. You won't be able to actually enter a romance with him until the Landsmeet though. The Crows will finally catch up to him while you're walking around the city. You'll need to kill them. Once it's over, Zevran will be free from the threat and he'll be open to love.
The important thing is that you don't push him too hard though. He will offer you an earing, but he will not appreciate it if you insist on it being a commitment of love. Just accept it and thank him. He'll then get a little distant while he mulls it over. Eventually he should be willing to admit his feelings and conclude the romance line.
Romance in Dragon Age Origins – Alistair
This is a fairly important romance, since it's possible for a human noble to become queen alongside Alistair through this path. Regardless, the actual romance isn't too tricky. It's not nearly as bad as the romance for Morrigan at least. Once again, you just can't make fun of him. He'll generally approve of "good" actions in the quest and you should gain approval pretty quickly. If you find that the romance has stalled, you can also follow his companion quest. All you have to do is have a fairly sad reunion with his sister. You have the opportunity to harden him through this quest too.
Once he's grown up a bit after his reunion, you should be able to fully fall in love. He'll offer you a rose and you can spend a night together in the tent.
This is a bit of a special relationship though, since the Landsmeet can end it. A normal Alistair will break up with you and honor his commitment to Queen Anora if you don't push for him to marry you and rule together. You need to pursue this path at the Landsmeet and convince him that they can't force the king to do anything that he's against. A hardened Alistair will be willing to continue the relationship as an affair during his rule.
Dragon Age Romance – Morrigan
A romance with Morrigan takes a little bit longer, although she's willing to sleep with the warden fairly early. Good characters will have a lot of trouble since Morrigan is usually at odds with the good characters. The problem is that Wynne, Alistair and Lelianna are all very useful. It's only later in the game that you can replace Wynne by teaching Morrigan healing power and replace Lelianna with Zevran. Sten is a decent stand in for Alistair, but that's about it for early options.
If you don't want to pick the more evil paths, then you will probably want to leave Morrigan behind for a fair number of the quests. It's either that or just try to keep her fairly placated until you can get the last few gifts. The thing with Morrigan is that all of her really important gifts are a little back loaded. Her companion quest grants a ton of approval and it should push you over the edge and into a real romance.
For Morrigan's gifts, you can generally focus on jewelry. The really important one is a golden mirror that you can buy in Orzammar. If you paid attention to her stories, you should remember a golden and bejewled mirror that she stole from a camp once but that Flemeth smashed to teach her a lesson on caution. Tell her that you remembered when you give her the mirror and you'll get a lot of apprvoal.
Doing her quest will net you a lot more though. When you go to the Circle Tower, you can find Enchanter Irving's office on the second floor. A special Black Grimoire is inside. Grab it. Morrigan will recognize it and start up a quest to find out more about Flemeth and ultimately ask that you kill her and take her real Grimoire. You'll need to kill Flemeth in dragon form, but it's not too bad for the advanced party you should have by that point.
This quest will give another big boost. After that Morrigan will ask you about your relationship and have some trouble coming to terms with her feelings. Make sure that you have a save ready or that you have a good quicksave for the camp. If you don't navigate it well then she'll break up with you. If you can convince her that it's a good thing, she'll give you a special ring.
That's about it though.
The Pearl – Threesome or Foursome
I'd be failing you guys if I didn't cover the option for a little extra action in The Pearl. Specifically, you can have the option for a threesome or even a foursome with Isabella.
She actually offers herself up when you talk to her after a little fight at The Pearl. She'll offer to teach you the Duelist specialization for a night together or by catching her cheating at the card game she's running. The option for extra partners will present itself for a few relationships. A hardened Alistair or Leliana will be willing to join you if you're already in a relationship. Zevran will also always offer to join as long as he's in the party, even if you aren't in a relationship.
Note that Morrigan isn't a libertine. If she's around and in love with you, she'll actively block the option for a night with Isabella and will be furious if you do it anyway.