Part fell off--can I fix it?

[PitronoftheArts]

Hi! When I was wiggling the header out of my pi when I heard something clatter on my desk...and realized it came from the pi. I've shown the part that came off in the image included: what does this do? Is the pi broken? Can I re-attach it?

[mahjongg]

first, its not "J1", as "J" labels are normally reserved for connectors, so instead of this (unlabeled) part J1 is the nearby microUSB connector.
It seems (educated guess) the decoupling cap for USB power, I'm guessing 10Volt 100uF 1206-size ceramic.
If you can solder it back on it should be fine, ceramic capacitors like this one are not "polarized" and so you can solder it on in both orientations, it does not matter which. If you cannot solder well, ask someone who does.
As its just a decoupling capacitor the zero might just work fine without it, if you use an adapter with short leads.

[PitronoftheArts]

Thank you so much! I'll just solder the part back on, in that case.

[rzusman]

Note that this is not your fault.
The part was clearly not soldered correctly, and you might want to look for other bad solder joints on the board.