A gazebo is a small open structure, not unlike a small, freestanding roof. They are usually found in public parks or gardens, and before the widespread use of the radio were very popular locations for speeches, debates, and local musicians.
Thanks to a 1994 Knights Of The Dinner Table story, reportedly based on true events, a gazebo is also some kind of creature in tabletop gaming culture which is completely mysterious, although the most knowledgeable seem to recall it is BIG, and probably very dangerous and is one of those specialty monsters DMs rarely use. In the comic, the main cast are given a description of the garden their characters enter, although thanks to their low real-life WIS and lack of real world awareness they believe the "gazebo" is some kind of monster, much to the frustration of their GM.
In the real world tale that the comic is supposedly based on, eventually, Eric is told his weapons are useless, so he does the only rational thing. He makes a run for it. Unfortunately, the Gazebo once awoken is as swift and deadly as it is cranky. Eric's paladin is killed without so much as a saving throw.
Fans of Dungeons & Dragons find a certain level of grim humor in this, because there's a very real D&D monster - the Mimic - which can, and does, disguise itself as objects before eating people dumb enough to treat it like one.
It's even in Munchkin. Although it's not too bad.
Other Cultural Influences
In addition to being spoofed and recreated a dozen times with varying animations accompanying the original skit, other Gazebo-related phenomena have appeared in the real world. In particular, there exists a twitter account (@KatsuconGazebo) for a gazebo placed in the Gaylord Convention Center in the National Harbor near DC. This twitter speaks as the vengeful spirit of the gazebo that is so often used as a photoshoot location by cosplayers attending the half dozen conventions that now use that venue. The running joke being that the gazebo is a capricious god that may grant cosplayers eternal fame, or may nom them.
Perhaps unsurprisingly, there are many eager supplicants for the latter.