Marriage was released alongside Valentine's Day 2014. To marry someone, you just have to enter the name of a user in the search box in the friends list (bottom left button with heart logo) and then click the marry button. You could also press on someone's username and then press "Marriage Proposal" from there.
Your spouse is officially called your "soul mate" (not to confuse with soulmate maps).
To divorce someone, click on their username from the friends menu and select "divorce."
It takes one hour to be able to marry again after divorcing someone.
Also, if you managed to becomes soulmates with someone without even bothering to befriend them first, you're stuck with them.
On Transformice's Trello page, a plan to improve the marriage interfaces was added on 14 April 2015.