If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of smoothness, working with topological manifolds (or even plain topological spaces) and homeomorphisms?
(Later edit: Argghh! I apologize, I forgot to specify: both curves do not have self-intersections and are homotopic.)
If your definition of "curve" is embeddings of say $[0,1]$ in a manifold, then the result follows in dimensions $n\geq 2$ by the isotopy extension theorem, together with the observation that the unit tangent bundle of a connected manifold is connected. In dimension $n=1$ the diffeomorphism would not always be isotopic to the identity, as the corresponding unit tangent bundle is not connected. The proof is primarily the isotopy extension theorem whose proof is an ODEs result and can be found in Hirsch's textbook on differential topology.
Some more details: there is precisely one isotopy-class of smoooth embedding of $[0,1]$ into a manifold if and only if the unit tangent bundle of that manifold is connected. This follows from the isotopy extension theorem in a standard way. For manifolds of dimension $n \geq 2$ their unit tangent bundles are connected if and only if the manifold is connected. For $n=1$ the unit tangent bundle is never connected.
Just to finish this thing off:
The following are equivalent:
A smooth manifold $M$ of dimension $n$ satisfies the property that for any two smooth homotopic 1-dimensional knots $K, K'$ in $M$, the pairs $(M, K), (M,K')$ are diffeomorphic .
$n\ne 3$.
The only nontrivial case is when $n=2$, it goes back to Dehn and Nielsen, see Andy's answer.
In the case that $M=\Sigma$ is a closed, oriented surface, this question has a rather complete answer; Farb and Margalit call it the "change of coordinates principle" in "The Primer on Mapping Class Groups." It says that given $\alpha, \beta$ two simple, homotopically non-trivial, non-separating (i.e. $\Sigma\backslash \alpha$ is connected) closed curves, there exists a diffeomorphism $f:\Sigma\rightarrow \Sigma$ such that $f(\alpha)=\beta.$ You can find this discussion starting on page 40 of the Primer. The proof is a pretty simple idea, you cut two copies of $\Sigma,$ one along $\alpha$ and one along $\beta,$ and then use the classification of surfaces to a obtain a diffeomorphism between these cut open surfaces (which are now surfaces with boundary) which is the identity on the boundary. Then you re-glue and obtain a diffeomorphism of $\Sigma$ which does what you want by construction. This can be extended to $k$-tuples of simple curves satisfying some simple homological hypotheses (again see the Primer).
