I apparently dropped a small piece of vinyl hose in the passenger side engine bay, which ended up resting on, and then melting on the pictured part when I turned the engine on. I removed the hose when I noticed the smoke. The smoke was from the hose, not the part. The car otherwise works fine, but I don't recognize this part. So, I don't know if I should be worried or plan to repair it.
It is an oxygen sensor (lambda sensor). Nothing to worry about.
Looks like a sensor on the exhaust manifold. Try to scrape off the plastic with a knife (to get rid of the smell), the sensor will be fine.