Let (X,x) be a pointed space. There is an action of π1(X,x) on πn(X,x) -- determined by considering πn(X,x)=πn-1(ΩxX,x), where ΩxX denotes the space of loops in X based at x, and x denotes the constant loop -- given simply by conjugation. We can speak unambiguously of πn(X), the free (i.e., not necessarily basepoint-preserving) homotopy group exactly when this action is trivial.
On an algebraic level I'm fine with this, but I'm having trouble envisioning how a homotopy class might be conjugated to a different homotopy class in this way. Besides my admittedly small collection of toy examples, my issue could also be that I'm mainly thinking about π1, in which case it might (???) be that the action is trivial. (I seem to recall that before learning about general homotopy theory, I heard a statement along the lines of "for path-connected spaces, you may as well ignore basepoints". Certainly the groups are all isomorphic, but I'm not certain whether there is a unique natural isomorphism.)
Also, are there (necessary and/or sufficient) conditions for when the π1 action on πn will or won't be trivial, and does this depend on n?