It's a half-hour OVA episode that was released in Japan as a "free gift" with the 16th volume of the manga. It contains two stories, one based on bonus chapter 88.5 from the manga's 15th volume and the other, interestingly, being original. (The anime has hitherto been painstakingly faithful to the manga, but hey.)
Next year, incidentally, Kamisama Kiss will get more OVA episodes, plus another stage musical. Yes, "another". It's not even the first.
FIRST STORY (12 minutes)
I've just read chapter 88 of the manga. It is indeed the same as its anime adaptation, although be warned that the OVA starts with a summary of season one. Spoilers galore. Don't watch this out of order. The important thing though is that it's funny, with a mysterious box and a bit at the end with drunken shapeshifters. It has Nanami being sensitive again, but understandably given the situation. I enjoyed it.
(The art's a bit sketchier than the anime, mind you. Sometimes a little, occasionally a lot. I like the energy that creates.)
SECOND STORY (17 minutes)
This is the richer of the two stories, I think. Partly that's due to its length, obviously, but it's also built more dynamically around its lead character.
By default, you see, Nanami's the normal one. She's the audience identification figure. She's the human. She's the one who has to put up with the daft antics of the supernatural boys. In other words, she tends to be the foil for everyone else's comedy. In this episode, though, she's decided that she needs to look after herself, or at least to demonstrate that she's capable of doing so. She can't just depend on Tomoe all the time. She can't be someone who's constantly being rescued from trouble. She's got to be tough, strong and a Jill of all trades!
In other words, she's being a goofball.
There's nothing wrong with the sentiment, obviously, although as so often there's a gulf between Nanami and Tomoe's perceptions. All this of course it's being played for comedy... and it turns out that Nanami can be hilarious and ridiculous in her own right. This is a Nanami-driven story with Nanami-driven jokes, which works brilliantly. She's always been funny, of course, but this story takes that to another level.
It's an onsen episode, but this is a shoujo series (i.e. for girls) and so it's not about fanservice. It takes them almost all the episode just to get to the onsen anyway. Meanwhile the intensity about touching a rock is funny. Psycho Nanami is awesome. Himemiko and Kotarou being lovey-dovey give us lots of nifty contrast with Nanami and Tomoe. I love the robotically inhuman way Yui Horie (Honda Touru in Fruits Basket) voices Himemiko, by the way. The silliness about hand-holding at one point was threatening to get a bit much, but Daichi is the past master of undercutting that kind of thing for laughs.
Is this OVA essential? No, obviously. It's a bonus extra. However it's funny and entertaining, not to mention briefly introducing someone at the end who looks as if he's going to be a villain in season two. Good stuff.