Difference between free homotopy and isotopy. Numer of non-isotopic curves. I realize that whenever I think of two simple closed curves in a surface being isotopic I actually think of them as being freely homotopic (intuitively). I am really confused now. So I have the following question.
(1): If two simple closed curves are freely homotopic are they isotopic and vise versa.?
(2)In a surface with nonempty boundary number of non-isotopic simple closed curves are finite or not? 
Any reference or hint will also be good enough. Counterexamples will be best. Thanks in advance.
PS: Two curves are isotopic means that there is an isotopy of the surface which takes one curve to another 
 A: If $S$ is a surface and $f_0,f_1\colon [0,1]\rightarrow S$ are two simple loops in $S$, we have the result that $f_0$ and $f_1$ are isotopic (with respect to $S$) if and only if they are freely homotopic. This is not true in higher dimensions of course because otherwise knot theory would be rather boring.
I believe a standard reference is Baer *. However there is a proof in English here kindly hosted for free by Benson Farb and Dan Margalit in their textbook-in-progress A primer on mapping class groups. This particular result can be found as proposition 1.10 on page 37 in Version 5.0.
For your second question, there exists a torus knot of type $(1,q)$, for every natural number $q\geq 0$, each of which is not isotopic with respect to the torus (they represent distinct elements in $\pi_1(T)$) and so there are an infinite number of isotopy classes of simple curves on the torus. Clearly for the sphere there is only one isotopy class because $\pi_1(S^2)=1$.
*R. Baer, Isotopien von Kurven auf orientierbaren, geshlossenen Fachen, Journal fur die Reine und Angewandte Mathematik, 159 (1928), 101-116.
