To be able to marry you have to wear Amulet of Mara. This unlocks dialogue with certain NPCs, so you can express your wish to marry him or her. Once that is done you can speak to Maramal at the Temple of Mara about arranging your wedding. The time to start a ceremony takes approximately 24 hours. Players have the option to engage in either a heterosixual or homosixual marriage. [1]
You can buy Amulet of Mara for 200 gold coins from Maramal. At first he can be found at the Bee and Barb, an inn of Riften town, and later in a Riften's Temple of Mara.
From: http://elderscrolls.wikia.com/wiki/Marriage