Have a Seat is originated from the hit show, Glee. It spread in popularity on tumblr shortly after episode 4 of the 3rd season, Pot 'O Gold. Even though it started after episode 4, it's origins are from episode 1 of the 3rd season. Within it, Blaine Anderson (for those not familiar), portrayed by Darren Criss, is a gay high schooler dating Kurt Hummel. Blaine goes to Dalton Academy and Kurt goes to McKinley High School. Kurt eventually convinces him to transfer and join the New Directions (their show choir or glee club, if you will).
This goes over well with most people, but not as much with the boys in the group, especially not Finn Hudson (played by Corey Monteith), the "leader" of the New Directions, step-brothers with Kurt Hummel, and boyfriend of the other "leader" of the club, Rachel Berry (Lea Michele). As soon as Blaine is introduced to the club, Finn informs him that it wasn't Dalton, they weren't the Warblers (Dalton's show choir who Blaine was lead singer of) and that they weren't interested in the bells and whistles or "the ball hogging". Needless to say, neither Kurt or Rachel were pleased to hear this from him, and by the 4th episode he'd only had 1 solo within the club, that he prepared himself.

Flash forward to the 4th episode. One of the members, Mercedes (Amber Riley) , left because of Rachel always being a spotlight hog, and another member, Tina (Jenna Ushkowitz), crying about it because she was one of the original members and it felt empty without her. Finn stands up and starts to give them a pep talk about how they're a family, they need to stay together, and how they shouldn't turn on each other. Then, Blaine stands up and agrees with Finn, adding how it's been a hard week and how they're a team so they should stay together. Finn then responds in his usual hypocritical fashion, telling him he "may be a big deal at Dalton, or whatever," but they don't wear blazers there, and he's trying to give a pep talk and tells him to "have a seat". After that Blaine responded with THE most relevant any needed to be said thing to Finn in the history of the show. "Didn't you just say something about us not turning on each other?" and sits down rightfully pissed off. Nonetheless, the rest of the episode, Finn is still a jerk to Blaine.

The theory of many fans of why Finn was being such a jerk about Blaine is that Finn has informed ability. He's good, but (debatably) Blaine is a much better singer and dancer (duh) than Finn and he's trying to protect his solos and position as "leader" though almost the whole fandom would agree that Blaine would be a way better leader because (once again, debatably) finn usually gets most of the male solos, and is generally hypocritical about solo distribution.

After the episode aired, on tumblr there were many posts involving conversations involving Finn and Blaine where Blaine is usually is interrupted by Finn where he's usually saying something along the lines of this isn't Dalton, we don't wear Blazers, and of the like, but they always end with Finn telling Blaine to have a seat.

example from tumblr- not mine:

Finn: Where are you going?
Blaine: I… was just going to the bathroom…? -
Finn: Yeah well you may have been allowed to go to the bathroom when you were a Warbler but this isn't Dalton, Blaine – so have a seat.
-
Finn: What are you doing?
Blaine: I was just singing my part of this song -
Finn: YEAH WELL WE DON'T WEAR UNIFORMS HERE, BLAINE.
Blaine: What do you even mea-
Finn: HAVE A SEAT.
-
Finn: You are never replacing Mercedes, Blaine!
Blaine: But that's not what I wan-
Rory: I am from Ireland!
Finn: WELCOME TO THE NEW DIRECTIONS, RORY. ALSO BLAINE? HAVE A SEAT.
-
Blaine: Excellent job, Rachel.
Rachel: Thank you, Blaine!
Finn: THIS ISN'T DALTON, BLAINE. WE DON'T STEAL EACH OTHER'S GIRLFRIENDS-
Blaine: But Dalton is an all-boys sch-
Finn: WE RESPECT EACH OTHER, BLAINE.
Blaine: But you cheated on Rach-
Finn: SO HAVE A SEAT.
-
Kurt: I love you, Blaine.
Blaine: I love you t-
Finn: YOU MAY BE A BIG DEAL AT DALTON OR WHATEVER, BLAINE. BUT WE DON'T WEAR BLAZERS HERE SO -
Kurt: Stfu.
Finn: Ok.
-
Finn: What are you doing?
Blaine: Er… Nothing?
Finn: GOOD. NOW HAVE A SEAT.
Blaine: I'm already sitting..
Finn: THEN HAVE A SEAT AGAIN.

Another common form of Have a Seat is GIFs involving the conversation with different text.

There are many examples on tumblr blogs devoted to Glee, especially Klaine.