The logic gate for which the output goes 'HIGH' or '1' only when an odd number of 'HIGH' or '1' are at its input, is

A logic gate providing a HIGH output exclusively when an odd number of inputs are HIGH can be identified by analyzing standard gate behaviors.

The Exclusive-OR (Ex-OR) gate is defined such that for two inputs $A$ and $B$: output is $1$ if either $A$ or $B$ is $1$ but not both, corresponding to an odd count (one) of HIGH inputs. For $A = 0$, $B = 0$, output is $0$ (even); $A = 0$, $B = 1$ or $A = 1$, $B = 0$, output is $1$ (odd); $A = 1$, $B = 1$, output is $0$ (even).

This characteristic of the Ex-OR gate aligns precisely with the given condition. OR gates output $1$ for any nonzero HIGH inputs; NAND outputs $0$ only when all inputs are HIGH; NOR outputs $1$ only when all inputs are LOW.

Final answer: $\boxed{C}$