Should the Dragonborn be interested in marriage, talk to Maramal at the Temple of Mara in Riften, though initially he will be found at the The Bee and Barb inn.
He will explain the procedure of how to get married. The people of Skyrim are rather practical about the whole matter. The Dragonborn needs to wear a Amulet of Mara (any one will do, it can be found, or bought from Maramal for 200 ). When then talking to suitable marriage candidates, a dialogue option for marriage will become available.
Note: There is a pretty well hidden "favour" system implemented in Skyrim. If you help NPCs they will have a positive disposition towards you. So for the above to work you will need to have made them "like" you enough first. E.g. by helping them with (all) their quests.
For a list of marriage candidates see Marriage Candidate NPCs. If the spouse agrees, then the marriage needs to be organized.
Speak to Maramal about arranging your wedding
Speak to Maramal about the wedding arrangements. The next day the spouse will be waiting at the altar at the Temple of Mara.
Attend your wedding ceremony
Attend the wedding at the Temple of Mara, it has to be on that very next day, or the quest fails. If all goes well the The Bond of Matrimony ring is received.
After the marriage, the Dragonborn must decide where they should live together. This may be at the spouses home, or at one of the homes the Dragonborn may have bought. It is possible to change homes at any time.