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 H0(M, ωn), where ω is a certain line bundle over M=compactified moduli stack of elliptic curves. There is non-trivial cohomology in
Hs(M,ωn)
for s>0. This comes in two flavors:
* free abelian group summands in H1(M,ωn) for n≤-10. These correspond by a kind of "Serre duality" to the usual modular forms in H0(M,ω-n-10).
* finite abelian groups (killed by multiplication by 24) in Hs(M,ωn) for arbitrarily large s.
There is a spectral sequence
Hs(M,ωt/2) ==> πt-s Tmf,
and the edge of the spectral sequence gives a map π2nTmf -> H0(M,ωn). 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.