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$n$ are H0(M, ωn)$H^0(M, \omega^n)$, where ω$\omega$ is a certain line bundle over M=compactified$M=$compactified moduli stack of elliptic curves. There is non-trivial cohomology in
Hs(M,ωn)$H^s(M,\omega^n)$
for s>0$s>0$. This comes in two flavors:
* free abelian group summands in H1(M,ωn)$H^1(M,\omega^n)$ for n≤-10$n\leq-10$. These correspond by a kind of "Serre duality" to the usual modular forms in H0(M,ω-n-10)$H^0(M,\omega^{-n-10})$.
* finite abelian groups (killed by multiplication by 24$24$) in Hs(M,ωn)$H^s(M,\omega^n)$ for arbitrarily large s$s$.
There is a spectral sequenceThere is a spectral sequence
Hs(M,ωt/2) ==> πt-s Tmf$H^s(M,\omega^{t/2}) \Rightarrow \pi_{t-s} Tmf$, alt text http://math.mit.edu/conferences/talbot/2007/tmfproc/EllipticSpectralSequence.pdf
and the edge of the spectral sequence gives a map π2nTmf -> H0(M,ωn)$\pi_{2n}Tmf \rightarrow H^0(M,\omega^n)$. 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$M$) is not the same as TMF (the 576$576$-periodic guy, which is sections on M'=stack$M'=$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.
