Spontaneous/Non-Spontaneous Reactions and Reversible Reactions
"Reversible reactions constitute a limiting case between spontaneous and non-spontaneous processes." So what does this mean? 


To me this sentence is meaningless gobbledygook and should be ignored.  That is, this sentence is unhelpful and trying to understand it will just make you more confused and less able to master chemistry.  Here is a short summary version of the points I make in the "Advanced discussion" below:


All reactions can be viewed as reversible from a mathematical standpoint, if the reacting system is "big enough".  Reactions that are highly spontaneous in the forward direction mean that at equilibrium, the number of "reactant" molecules will be very very small (but not zero!).  
Many times, "big enough" would mean astronomically big, such as bigger than the Earth.  In these cases, reactions can be regarded as "irreversible", although there is no strict, universally agreed upon boundary between "reversible" and "irreversible".
Whether a reaction is spontaneous isn't that much related to whether it is "reversible".  Don't worry about understanding the connection between these concepts until you have understood each concept independently.


Advanced discussion.

The meaning of spontaneous

Spontaneous processes have negative Gibbs free energy changes ($dG &lt; 0$) and also result in positive entropy changes in the universe ($dS_{univ} &gt; 0$).  However, $dG$ is a function of state.
For chemical reactions, this means that $dG$ depends on the concentration of the reactants and products, i.e.

$$dG = dG^\circ + RT \ln Q$$

where $Q$ is the reaction quotient.

Thus, if we start off with all "product" and no reactant, then the "reverse" direction of the reaction will be spontaneous.  But if we start off with all reactants and no product, then the forward direction is spontaneous.

The meaning of reversible

The short answer is that for chemical reactions, "reversible" does not have a precise thermodynamic definition.  

$$dG = dG^\circ + RT \ln Q = -RT \ln K + RT \ln Q = RT \ln \frac{Q}{K}$$

where $K$ is the equilibrium constant.  People will usually call reactions with a "large" equilibrium constant irreversible and those with a "small" equilibrium constant reversible.  But the exact definition of "large" and "small" isn't well-defined.  Many people might say $\frac{Q}{K} \leq .01$ represents an "irreversible" reaction.  For chemical reactions with very large equilibrium constants, the size of the system required to ensure that an equilibrium can be reached can get very large, as explained here.