We construct the "spectral" decomposition of the sets $\overline{Per\,f},$ $\omega(f)=\cup\omega(x)$ and $\Omega(f)$ for a continuous map $f:[0,1]\rightarrow [0,1]$. Several corollaries are obtained; the main ones describe the generic properties of $f$-invariant measures, the structure of the set $\Omega(f)\setminus \overline{Per\,f}$ and the generic limit behavior of an orbit for maps without wandering intervals.  The "spectral" decomposition for piecewise-monotone maps is deduced from the Decomposition Theorem.  Finally we explain how to extend the results of the present paper for a continuous map of a one-dimensional branched manifold into itself.