It's all to do with the chords relative to the key. Unless you're doing something really weirdly modal, most stuff will at some point live in the 3-chord trick, the tonic, third and fifth above it. Wherever the key, the relationship to the tonic remains the same.
So if the tonic is C, the third is E and the fifth G (C=1, D=2, E=3, F=4, G=5, A=6, B=7). The chord of C is C-E-G-C, E is E-G-B-E, G is G-B-D-G. Shifting the nut using a capo changes the key upwards a semitone per fret but the shape of the chords remains the same. It's why we talk about diminished fifth rather than GDim, you can transpose easily.
The downside is that not every key sounds the same. Some are brighter, others sadder, it's the price you pay for shifting a tune from the key it was intended to be in to one you can sing it in.

Another approach to it is using solfege, so we're talking about Doh-Mi-Sol and the chords thereon in whatever key. Doh is the tonic, Mi the third, Sol the fifth. By default on the key of CMaj, but not necessarily. Solfege describes the relative intervals, which is important when using modal keys (ie using the notes of one key but with a scale running from a note other than the tonic).