Yeah, I would say that it goes on the 2nd fret.
The familiar open position chord shapes (those that use open strings) still work, but that C major chord in reality becomes a D major. To play an actual C chord with a capo, there's still the root at the 3rd fret 5th string. You just need to find the others. The next most common barre chord is shown below.

These are merely the usual string notes, not the actual notes that ring when capoed at the 2nd fret.
E---3
B---5
G---5
D---5
A---3
E----
Now whether the song you're learning truly intends the chord to be a C or merely the usual C chord fingering moved up 2 frets I dunno. And yeah...usually a song (played with the exact same chord shapes) will sound totally different compared to without a capo.