Q:

Meaning of throwdown in the context

Saw this on Instagram. 

We've invited Google to WeWork Fulton Market for a hamburger
  throwdown. We're pretty confident.

What does throwdown mean here?

A:

"Throwdown" is the nominal version of "throw down" as in "throw down the gauntlet" meaning to challenge someone to a fight or contest, originally of arms.
Here is an excerpt from an article by Elizabeth Harrison on the History Channel website.

Today the phrase “throw down the gauntlet” means to challenge or confront someone, but in its earliest use it wasn’t meant as a metaphor, but was a physical action intended to issue a formal challenge to a duel. The word itself comes from the French word “gantelet,” and referred to the heavy, armored gloves worn by medieval knights. In an age when chivalry and personal honor were paramount, throwing a gauntlet at the feet of an enemy or opponent was considered a grave insult that could only be answered with personal combat, and the offended party was expected to “take up the gauntlet” to acknowledge and accept the challenge.

So throwdown means a confrontation or challenge of some sort. A "hamburger throwdown" would be a contest to see who could make the best hamburgers (cf. other food challenges, e.g., a "chili cook-off" etc.)

