Is mass the only factor which affects Newton's Third Law? I've had a hard time understanding Newton's third law. From my textbooks, it can be inferred that the reaction of the object differs based on its mass. For example, if skater A pushes skater B, the lighter skater will accelerate further. However, based on readings on this website, other factors such as friction are being discussed in for example, a person pushing a table.
This has confused me, and it would be of great help if someone were to help me truly understand this law.
 A: 
For example, if skater A pushes skater B, the lighter skater will accelerate further.

Good enough so far.
But you should be careful about the reasoning. Newton's third law says that when skater A pushes on skater B, there is an equal force applied by skater B to skater A (Mass isn't actually a consideration in Newton's third law). It's Newton's second law that tells you this will result in the lighter skater experiencing more acceleration.

However, based on readings on this website, other factors such as friction are being discussed in for example, a person pushing a table.

Friction doesn't change Newton's laws. It just introduces a third object (the Earth) into the system.
Newton's third law says, if I push on the table, there is an equal force pushing back on me. Friction means that the table will also be pushing on the Earth beneath it, and (because of Newton's third law) the Earth will also push back on the table. Newton's third law hasn't changed, we just have to consider two interactions where it applies, instead of one.
A: 
From my textbooks, it can be inferred that the reaction of the object
differs based on its mass. For example, if skater A pushes skater B,
the lighter skater will accelerate further.

Mass comes into play when applying Newton's Second Law, not Newton's Third Law. Newton's third law only states that when skater A applies a force to skater B skater B applies an equal and opposite force on skater A. What the consequences are of these forces on each skater will depend on the application of Newton's second law to each skater individually where the applicable force on each is the net force applied to each, or
$$a=\frac{F_{net}}{M}$$
For example, let the force that each skater applies to the other be $F$. Suppose when skater A applies a force $F$ on skater B skater B happens to be holding on to a rigid hand rail that surrounds the skating rink. If the hand rail applies a force on skater B equal and opposite the force that skater A applies to skater B, the net force on skater B is zero and skater B does not accelerate.
Let's say skater A is simply standing on the ice with nothing else restraining skater A than the friction force of the ice. Then if the reaction force $F$ of skater B on skater A is greater than the opposing friction force of the ice, skater A will experience a net force of
$$F_{net}=F-F_{ice}$$
And skater A will accelerate backwards away from skater B according to
$$a_{A}=\frac{F-F_{ice}}{M_{A}}$$
Hope this helps.
