A "Boston marriage" is a long-term intimate friendship between two women who usually live together. The term probably came to English as a reference to Henry James' 1886 novel The Bostonians, which focuses on Olive Chancellor, a Boston woman who we would now call a feminist; her conservative Southern cousin Basil Ransom; and the younger woman Verena Tarrant who is staying with Olive and working toward becoming a feminist speaker. Basil is attracted to Verena, if not her ideas, and the novel is a sort of tug-of-war between Basil and Olive over Verena. James may have modeled Olive and Verena on the writer Sarah Orne Jewett and poet Annie Adams Fields, or perhaps on his own sister Alice James and her relationship with Katharine Loring.
At the time, living with another woman was a (somewhat) socially acceptable alternative to marriage that preserved a lot more independence than continuing to live with blood family members. (Living alone was just not something that nice women did at the time; many people probably viewed two women living together as each chaperoning the other.) This kind of cohabitation was more common among educated women who could support themselves in some career.
However, the idea of "Boston marriage" goes a lot farther than just having a roommate; it's more of an intentional community or fictive kinship. Theoretically, such relationships are platonic, since the label originated in a time when women were commonly assumed to have little sex drive; however, it is likely that some women in Boston marriages in the past were lesbians. Therefore, the term has developed a connotation of euphemism. (And since the state of Massachusetts legalized same-sex marriages in 2004, the term is now occasionally re-interpreted as an official marriage performable in that capital city.)
"In my day, TV stars weren't allowed to say 'booby', 'tushy', 'burp', 'fanny-burp', 'water closet', 'underpants', 'dingle-dangle', 'Boston marriage', 'LBJ', 'Titicaca', or 'frontlumps'!"
--Grandpa Simpson of The Simpsons
Boston Marriage is also a 1999 David Mamet play, set in the late 1800s. It concerns Anna and her younger companion Claire, and the conflict over their other relationships; Anna is the mistress of a wealthy man and Claire is falling in love with a naive young girl. Reviews describe it as an Oscar Wilde-esque satire of society and people's relationships.