Here's what Wikipedia says about Black Day:
"Black Day (April 14) is a South Korean informal tradition for single people (a.k.a. Unit Solo/Solo Regiment, lee: 솔로부대) to get together and eat Jajangmyeon (noodles with black bean sauce), sometimes a white sauce is mixed for those who did not celebrate White Day.And what did I do to celebrate Black Day? I watched Revolutionary Road. If there was ever a movie that reminded me more of how I felt in the short time I was married, it was that one. About five minutes into the movie, I was convinced the wife was going to stick her head in the oven by the end. While that may not have been what happened, it certainly communicates the tone of the movie. I can't imagine a better way to celebrate singleness on Black Day than by watching something that made me feel so incredibly happy with my decisions!
The idea is that those who did not give or receive gifts on Valentine's Day (February 14) or White Day (March 14) can get together and eat Jajangmyeon (짜장면), white Korean noodles with black bean sauce (hence the name), to commiserate their singledom."