Definition : Let G be a group and g and x elements in it. Then the conjugate of g by x, often written gx, is xgx-1. Can also be defined as x-1gx - it doesn't really make much of a difference.
Although everything we can prove about conjugation applies generally, a nice way of looking at conjugation comes from the case of G acting on a set Ω. So here x and g are actions, be they rotations or permutations or whatever, and we see that gx is the action of doing x, doing g, and then undoing x. So we're moving whatever we're acting on into some state, doing g, and then moving it back into what would be the original state except had we not done g in between. So anything (any element of Ω) which g leaves alone in that moved state is going to return to where it was unchanged, and in general we can see that g and gx are going to look very much alike.
The basic point is that gx "does the same thing" as g, but it does it to different elements of Ω - in particular to those which x puts (in the finite case, permutes) into the positions which g acts on. So (in the group of linear transformations of Euclidean space) the conjugate of a rotation will be a rotation, and (in the group of moves on a Rubik's cube) the conjugate of a double corner swap will be a double corner swap.
This is a consequence of the more abstract fact that conjugation by a fixed element x is an automorphism, i.e. an isomorphism from G onto itself. The proof is straightforward from the definition.
Much more can be said about conjugation, but I'll constrain this write-up to the following. Firstly, note that g1 = 1g1-1 = g and gxy xyg(xy)-1 = xygy-1x-1 = (gy)x, so conjugation satisfies the axioms for an action of G on itself. And hence G partitions into orbits - the conjugacy classes of G. And by the theorem in coset space, we have that the conjugacy class of which an element g is a member, Orb(g), is isomorphic to (G : Stab(g)) - the coset space of Stab(g) in G. With the action of conjugation, the stabilizer Stab(g) is also known as the centralizer of g, and consists of the elements of G which commute with g (since xgx-1 = g <==> xg = gx).
So we have that the structure of a conjugacy class is the same as that of the set of cosets of the elements which commute with any fixed element of the conjugacy class, and hence also that those coset spaces are all isomorphic to each other - and in particular have the same size. And so by Lagrange's theorem, we have the (not immediately obvious) result that the centralizers of elements of a conjugacy class have the same size(/cardinality - all this applies to infinite groups as well as finite ones).