Two elements $x,y$ of a group $G$ are conjugate if there is some $h\in G$ such that $hx{h}^{-1}=y$.

Conjugacy as "changing the worldview"

Conjugating by $h$ is equivalent to "viewing the world through $h$'s eyes". This is most easily demonstrated in the symmetric group, where it is a fact that if $$\sigma =(a_{11}{a}_{12}\dots a_{1n_1})(a_{21}\dots a_{2n_2})\dots (a_{k1}{a}_{k2}\dots a_{k{n}_k})$$ and $\tau \in S_n$, then $$\tau \sigma {\tau }^{-1}=\left(\tau \right(a_{11}\left)\tau \right(a_{12})\dots \tau (a_{1n_1}\left)\right)\left(\tau \right(a_{21})\dots \tau (a_{2n_2}\left)\right)\dots \left(\tau \right(a_{k1}\left)\tau \right(a_{k2})\dots \tau (a_{k{n}_k}\left)\right)$$

That is, conjugating by $\tau$ has "caused us to view $\sigma$ from the point of view of $\tau$".

Similarly, in the dihedral group $D_{2n}$ on $n$ vertices, conjugation of the rotation by a reflection yields the inverse of the rotation: it is "the rotation, but viewed as acting on the reflected polygon". Equivalently, if the polygon is sitting on a glass table, conjugating the rotation by a reflection makes the rotation act "as if we had moved our head under the table to look upwards first".

In general, if $G$ is a group which acts as (some of) the symmetries of a certain object $X$ ^[1] then conjugation of $g\in G$ by $h\in G$ produces a symmetry $hg{h}^{-1}$ which acts in the same way as $g$ does, but on a copy of $X$ which has already been permuted by $h$.

Closure under conjugation

If a subgroup $H$ of $G$ is closed under conjugation by elements of $G$, then $H$ is a normal subgroup. The concept of a normal subgroup is extremely important in group theory.

1. ^{^︎}

   Which we can always view as being the case.