There are a handful of specific non-player characters in the game whom you are allowed to marry without restrictions to race or gender. Approximately 28 female characters and 34 male characters are available to marry, each represented by a wide range of different races. NPCs that can be married generally offer you a quest to do to win their affections. When wearing the Amulet of Mara, you get the dialogue option to propose to them.
There are actually a number of different Amulets of Mara in the game, and any one of them works for the purposes of getting married. Maramel sells them for 200 gold at both the Bee & Bard and the Temple of Mara in Riften. An Amulet of Mara can also be found in a treasure chest in Bleak Falls Barrow, in a tent on a beach north of Dawnstar and by stealing from Kematu in Swindler's Den. These Amulets can also appear randomly as loot during the Book of Love quest given by Dinya Balu or in a chest in the final room of Reachcliff Cave.
Your relationship with your chosen partner can be improved by talking to him or by hiring him in certain cases. The main way to improve the relationship is to perform a quest your partner gives you. Once his disposition to you is high enough, he is likely to agree when you propose to him while wearing the Amulet of Mara.