  We study open-closed orbifold Gromov-Witten invariants of 3-dimensional
Calabi-Yau smooth toric Deligne-Mumford (DM) stacks (with possibly non-trivial
generic stabilizers and semi-projective coarse moduli spaces) relative to
Lagrangian branes of Aganagic-Vafa type. We present foundational materials of
enumerative geometry of stable holomorphic maps from bordered orbifold Riemann
surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack with boundaries
mapped into a Aganagic-Vafa brane. All genus open-closed Gromov-Witten
invariants are defined by torus localization and depend on the choice of a
framing which is an integer. We also provide another definition of all genus
open-closed Gromov-Witten invariants based on algebraic relative orbifold
Gromov-Witten theory; this generalizes the definition in Li-Liu-Liu-Zhou
[arXiv:math/0408426] for smooth toric Calabi-Yau 3-folds. When the toric DM
stack a toric Calabi-Yau 3-orbifold (i.e. when the generic stabilizer is
trivial), we define generating functions of open-closed Gromov-Witten
invariants or arbitrary genus $g$ and number $h$ of boundary circles; it takes
values in the Chen-Ruan orbifold cohomology of the classifying space of a
finite cyclic group of order $m$. We prove an open mirror theorem which relates
the generating function of orbifold disk invariants to Abel-Jacobi maps of the
mirror curve of the toric Calabi-Yau 3-orbifold. This generalizes a conjecture
by Aganagic-Vafa [arXiv:hep-th/0012041] and Aganagic-Klemm-Vafa
[arXiv:hep-th/0105045] (proved in full generality by the first and the second
authors in [arXiv:1103.0693]) on the disk potential of a smooth semi-projective
toric Calabi-Yau 3-fold.
