If you know about classical modular forms, and you want to understand what they have to do with tmf, it is good to contemplate something more classical: the "derived functors of modular forms".
Modular forms of weight $n$ are $H^0(M, \omega^n)$, where $\omega$ is a certain line bundle over $M=$compactified moduli stack of elliptic curves. There is non-trivial cohomology in
$H^s(M,\omega^n)$
for $s>0$. This comes in two flavors:
* free abelian group summands in $H^1(M,\omega^n)$ for $n\leq-10$. These correspond by a kind of "Serre duality" to the usual modular forms in $H^0(M,\omega^{-n-10})$.
* finite abelian groups (killed by multiplication by $24$) in $H^s(M,\omega^n)$ for arbitrarily large $s$.
$H^s(M,\omega^{t/2}) \Rightarrow \pi_{t-s} Tmf$,
and the edge of the spectral sequence gives a map $\pi_{2n}Tmf \rightarrow H^0(M,\omega^n)$. This is essentially the map Chris describes in his answer.
Slightly confusingly, the gadget I've called Tmf (which is the global sections of a sheaf of spectra over $M$) is not the same as TMF (the $576$-periodic guy, which is sections on $M'=$stack of smooth elliptic curves), or tmf (the connective cover of Tmf, which I don't believe is known to be sections on anything).
I recommend Goerss's Bourbaki talk for learning more about this, especially section 4.6.
