There are a few people you can marry some in small villages or mills others in the city. You can't marry just anyone, and you can only marry once. The first person that will marry you is the Riverwood trader girl, as you do the quest that is related to her, but there are others. Largely it doesn't matter who you marry, the benefits are all the same. She will cook a meal a day for you and give you the proceeds from the shop she sets up (100gp) you can also trade with her. If you marry one of the available NPC's that will adventure with you then they can still adventure with you when they are your wife.
All you need is the Amulet of Mara, and a little instruction from the priest and you are good to go and find some woman, man, Argonian, Dark Elf, High Elf that is willing to be your husband/wife or what have you.