This year, the Lunar Festival runs from Sunday, Jan. 27 to Saturday, Feb. 9. (As always, check your game calendar for specific start and stop times for your server.) On a very important note, this year cuts the length of the holiday from three to two weeks, so if you're a new player or you're starting a fresh character, you won't have as much time as you usually do to get everything done. I think Blizzard limited it so as not to clash with Love Is In the Air, which begins Sunday, Feb. 10.
If you're new to the Lunar Festival, the holiday mostly revolves around involve talking to a number of elders in order to honor them, receiving Coins of Ancestry in return, and then using the coins to purchase holiday items, including clothing and pets. The To Honor One's Elders meta-achievement is also a requirement for What A Long, Strange Trip It's Been, so if you're still after a Violet Proto-Drake, make sure you get this done.
The Lunar Festival is definitely not one of Azeroth's more difficult holidays, but it's time-consuming if you need all the NPCs. However, it's also a good time to do other travel-intensive achievements like World Explorer.