The act of making someone believe a joke, statement, etc. is true, while it is in fact completely fake.
Usually the person being januaried will try very hard to figure out what the person meant, making them look like a complete and utter idiot, as they meant nothing. Added people who are in on the january will add to the hilarity.
The most common use of the term is through the "January Joke":
Person 1: Yo, what month comes after June?
Person 2: July?
1: Are Julying to me?
2: Ha ha ha very funny.
1: Ok listen to this one: What month comes after December?
1: Are you Januarying to me? *Everone "in" on the joke starts laughing*
2: (Embarrassed) What? I don't get it. What do you mean?
When someone is found out:
Person 1: Dude I brought home like 20 chicks last night man!
Person 2: Yeah right, you're januarying me...
Person 1: Damn you got me.