What does having a bar on a manifold mean? If $M$ is a manifold then what is denoted as $\overline{M}$?
I am guessing that it means a reversal of orientation. 
Related to the above I would like to understand the following construction of a $d$-dimensional Riemannian category that I recently came across:
Objects are $(d−1)$ Riemannian manifolds. Morphisms between two oriented $(d − 1)$-dimensional Riemannian manifolds $N_1$ and $N_2$ are oriented $d$-dimensional Riemannian manifolds $M$, such that $\partial M = N_1 ⊔ \overline{N_2}$. The orientation on all three manifolds should naturally agree, and the metric on $M$ agrees with the metric on $N_1$ and $N_2$ on a collar of the boundary. The composition is the gluing of such Riemannian cobordisms.
I would like to know more about this intuition of defining a morphism between two manifolds as another manifold itself! One is used to thinking of morphisms as maps. 
 A: You've simply encountered the definition of a topological quantum field theory (in the formulation gave by M. Atiyah). I can provide you lots of documentation about that, but I'm a little hurried now... I'll edit this post during the day. For the moment I'm copypasting this:
Topological Quantum Field Theory, or TQFT for short, is a notion which originally arose from ideas by E. Witten in quantum physics (SUSY-Quantum Mechanics). Since then it has developed considerably in a number of directions, and in particular has had a pervasive influence in algebraic topology, by means of its capacity of computing algebraic invariants of a suitable class of manifolds, using so-called "cobordisms" between closed manifolds (smooth manifolds having them as boundaries).
Atiyah gave a first axiomatization of Witten's theory in his paper [1]; I'll try to focus on the illuminating analogy (duality) beween Atiyah's axioms  for TQFT and the Eilenberg-Steenrod axioms for homology as in [2].
Atiyah's approach is implicitly categorical; Jacob Lurie recovered this approach and considerably strengthened it in his paper [3]: by now we define an "n-dimensional, k-valued Topological Quantum Field Theory"as a monoidal functor
between the category of n-cobordisms and that of finite dimensional k-vector spaces.
If you have time and spirit to do so, I warmly recommend to give a glance to a "simplified n-categorical formulation" of TQFTs given by B. Toen 
in his [4].
References.
[1] Atiyah, Michael (1988), "Topological quantum field theories", Publications Mathématiques de l'IHÉS 68 (68): 175–186
[2] Samuel Eilenberg, Norman E. Steenrod, Axiomatic approach to homology theory, Proc. Nat. Acad. Sci. U. S. A. 31, (1945). 117–120.
[3] Lurie, Jacob, On the Classification of Topological Field Theories
[4] Bertrand Toen, Higher categorical structures in TQFT. Lecture in MPI-Bonn, June 2000.
A: 
I would like to know more about this intuition of defining a morphism between two manifolds as another manifold itself! One is used to thinking of morphisms as maps.

One intuition is from physics: one thinks of $n$-manifolds as "space," and of $n+1$-dimensional cobordisms between them as "time evolution." A nice introduction to these ideas is Baez's Physics, Topology, Logic and Computation: a Rosetta Stone. 
