How can we have massive states of strings and CFT on the string worldsheet at the same time? Ok, so we can have conformal invariance on a string world sheet. However, it is well known that to preserve conformal symmetry we require states to be massless. So how is it that string theories incorporate CFT but allows massive states?
Is it because the CFT is on the worldsheet and therefore applies to the worldsheet coordinate X (X is treated as the field) - however the physical states arise from the the creation/annihilation operators that we get from X? Therefore the CFT doesn't actually act on the states (massive or massless) but instead it acts on the field X.  
 A: I think you've basically identified the answer in your question: the massive states have a mass in spacetime, while the worldsheet theory is a CFT. Operators on the string worldsheet correspond to fields in spacetime, and the correspondence is roughly that high-dimension worldsheet operators correspond to heavy fields in spacetime. For instance, if you turn on a background field $M_{\mu\nu}(X)$ sourcing the operator $\partial X^\mu \partial X^\nu$ on the worldsheet, its symmetric and antisymmetric parts correspond to the massless spacetime fields $g_{\mu\nu}$ (the metric) and $B_{\mu\nu}$ (the antisymmetric $B$-field) in spacetime. More complicated operators on the worldsheet (say, $\left(\partial X\right)^{6}$) would give you less familiar massive fields in spacetime: these are string modes.
(This is, in a sense, the prototype for a similar issue in AdS/CFT, where sources for high-dimension CFT operators turn out to give rise to heavy fields in the Anti de Sitter dual in one dimension higher.)
