Classes who can perform the Wedding Ceremony include:
Clergy Lords, Templars, Prelates, Friars, Clerics and Shamans
Characters who want to marry each other must first find a Clergy character capable of performing the action 'Wedding Ceremony'. They must then get the agreement of that character to perform the ceremony, and ensure that their chosen officiant is supplied with Linen, Florins and a single Cut Gem. This means you need to a) be near a market that sells these things, b) been very lucky with action and scenario rewards, or c) own buildings that produce these materials in a sustainable production chain. Giving these items to your officiant is also an act of trust, it is possible for them to decide to take your goods and run, as long as they don't mind being trashed in the forums. Look for an officiant with a good reputation.
Finally the Clergy officiant spends the goods, some magic, and a rather lot of Action Points to create a Proposal Scenario. It is their job to correctly select the two characters that wish to marry. The two players the officiant selects can then agree to or reject the marriage proposal (proposed by the officiant). If both players agree, they become married and gain all the in-game benefits of spousal union. This benefit is primarily a powerful importance-sharing bond and the display that your characters are officially married. This also allows the two of you to form a 'Family' group together.