Basics
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.
Leliana
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.