NALT is a term coined by sex advice columnist and progressive activist Dan Savage that stands for "Not All Like That." It refers to those Christians (or presumably members of other groups) who make whispered apologies for others of their ilk with whom they say they disagree, and who they say do not speak for them. Their protest is that although, for instance, fundamentalist Christians who consider homosexuality an abomination and a sin gather a lot of attention, that not ALL Christians actually feel that way.
(Savage's contention is that these people should spend less time telling HIM that "not all Christians" think that way, and instead tell the pope, or tell people like Tony Perkins, who say they're acting as mouthpieces of the faithful.)
"Christians believe that allowing same-sex couples to marry would destroy the very fabric of society." A NALT Christian is someone who disagrees with this, and quietly explains that, hey, "We're Not All Like That."