How good are you at doing things when you're put on the spot?
How good are you at doing something when you're put on the spot?
Is the second one grammatically incorrect? What's the difference in the meaning between the sentences?
Both are grammatical. The difference in meaning is extremely small.
I would say that the difference is in focus. If you say "things", this is general, and there is no particular focus. If you say "something", you are choosing to use a singular (you could also say "a thing") and in some way you are putting a focus on that single thing. But as I say, the difference is very slight.