An important theorem by Beilinson, describing the bounded derived category of coherent sheaves on P^n, is extended to every weighted projective space P(w). To this purpose we consider, instead of the usual category of coherent sheaves on P(w), a suitable category of graded coherent sheaves. The weighted version of Beilinson's theorem is then applied to prove a structure theorem for good birational weighted canonical projections of surfaces of general type. This is a generalization of a theorem by Catanese and Schreyer (who treated the case of projections into P^3), and is mainly interesting for irregular surfaces, since in the regular case a similar but simpler result (due to Catanese) was already known. The result is then used to study a family of surfaces with numerical invariants p_g=q=2, K^2=4, projected into P(1,1,2,3).