First off, Visit Riften. In Riften theres a temple called Temple Of Mara, While inside you must talk to Maramal.


After finding Maramal talk to him about marriage for him to offer you an Amulet of Mara for 200 septims.


You must wear the amulet for the marriage dialogue to appear. Some of those you wish to wed may give you a little quest to prove your worth before you can marry them.
For those wondering, I believe the console version doesnt support same sex marriage.
If you own the PC version theres some input you can put into the cheat window to marry any NPC.
The List of available NPCs available for marriage:

Female:
Aela the Huntress
Aeri
Anwen
Avrusa Sarethi
Borgakh the Steel Heart
Brelyna Maryon
Camilla Valerius
Dravynea the Stoneweaver
Ghorza gra-Bagol
Gilfre
Grelka
Iona
Jenassa
Jordis the Sword-Maiden
Mjoll the Lioness
Muiri
Njada Stonearm
Orla
Ria
Senna
Shahvee
Sylgja
Taarie
Temba Wide-Arm
Uthgerd the Unbroken
Viola Giordano
Ysolda

Male:
Ainethach
Angrenor Once-Honored
Argis the Bulwark
Athis
Balimund
Belrand
Benor
Calder
Cosnach
Derkeethus
Erik the Slayer
Farkas
Filnjar
Gat gro-Shargakh
Ghorbash the Iron Hand
Marcurio
Moth gro-Bagol
Octieve San
Odfel
Omluag
Onmund
Pavo Attius
Perth
Quintus Navale
Revyn Sadri
Roggi Knot-Beard
Romlyn Dreth
Scouts-Many-Marshes
Sondas Drenim
Sorex Vinius
Stenvar
Torvar
Vilkas
Vorstag
Wilhelm