Correct dialogue choices initiate romances while increased approval ratings and personal quest completions advance them. There are at least two opportunities to initiate a romance with each character, although these opportunities are not always the same for player Wardens of both sexes. There is at least one opportunity early on in the game. Another opportunity occurs after that particular character's quest has been completed. Some characters are open to casual sex and thus have lower requirements for physical intimacy (Morrigan, Zevran), while others are only looking for serious, long term commitments (Leliana, Alistair) and thus require more investment. You can tell that a romance is initiated if their rating is Interested/Care/Adore/Love instead of Neutral/Warm/Friendly