It used to be a church, at least. What you can't quite see in this photo, but which is much clearer from the other side, is that the facade is the only part of the structure still standing. The rest, erected in 1870 by St. Ann's Roman Catholic parish (the church became St. Ann's Armenian Catholic Cathedral in 1983), was demolished in 2005 to make way for the 26-story NYU dorm standing behind it, the tallest building in the East Village. (Not long after the dorm was put up, the City Council approved a rezoning plan limiting building heights in the area to about 12 stories.)
This was actually an example of history repeating itself, to a degree. The extant facade dates to 1847, when the original church on the site was built by the 12th Street Baptist Church. (The building later became a synagogue, home to Congregation Emanu-El.) But in 1870, leaving the facade in place, St. Ann's tore down the rest of the structure and replaced the sanctuary with a new one, which was then in turn replaced by the dorm some 135 years later.
As you might imagine, many neighbors and preservation-minded people were upset with the destruction of the church's body and the subsequent construction of such a bland-looking behemoth in its place. I like the overall effect though, specifically the weirdness of there being a 19th-century stone church facade standing, for no apparent reason, in front of an insipid 21st-century apartment tower. The facade is now effectively a piece of public sculpture and something of an architectural folly (meaning "a whimsical or extravagant structure built to serve as a conversation piece, lend interest to a view, commemorate a person or event, etc."), which I think probably encourages more passersby to interact with it than ever did with the church when it was still intact.