The effectiveness of an air conditioner in dehumidifying

Is an A/C almost as good as an actual dehumidifier machine itself, or is it only a small fraction as good when it comes to removing moisture from air?

If the A/C is sized correctly, it will be much more effective and efficient than a dehumidifier. First of all, it will remove humidity from the whole house, where a dehumidifier is only good for a small area.
Secondly, it will reject its waste heat outside, instead of inside, where it will have to be removed (by an A/C, presumably).

Gotcha. I’m thinking the HVAC system that lies in the center of a whole house/apartment. Since it is running air over cold coils, it condenses the water out the same way right? (I’m wondering about the efficiency because I haven’t found the cold air from HVAC to feel as dry as that which comes out of a dehumidifier)

Well, a dehumidifier is just an A/C, after all.
What temperature is the air coming out of the A/C? It should be around 50-60°F. If it’s much warmer, than the A/C is either not working correctly, or it’s undersized, or the fan is moving too much air past the coils.
How long is a typical A/C cycle? It should be at least 10 minutes or so. If the unit is “short-cycling,” it is oversized, and won’t extract as much water as it should.

An air conditioner removes water and cools the interior. A dehumidifier removes the water but does not lower the temperature (since the heat is not pumped outside). If you house if a very humid 60 F, the air conditioner isn’t going to help much. In general, this isn’t much of a problem since humidity is typically more of a problem when the temperature is much higher, but it can be.

As to which would do a better job of removing water from the air, it really depends on the relative sizes of the two units,

Gotcha. I am in the 2nd-floor apartment of a three-story building. Am I generally only dehumidifying my own air, (by turning on my central HVAC), or am I also drawing air from downstairs into upstairs and dehumidifying air from the below unit as well?

In an apartment building each air handler should be separate and only cycle the air from your unit.

Both will Dehumidify, but it depends on what you want. Do you want it hot or cool? Max dehumidifying would be to run both, the a/c will, in general, be more powerful then the dehumidifier so the temperature should fall running both.

This will depend on whether we’re talking about a single-room A/C unit, or central A/C. A single-room A/C unit and a dehumidifier both have roughly the same upper bound on power consumption, i.e. what can be had from a standard 110V outlet; one would expect them to have roughly the same ability to dehumidify. Central A/C is typically powered by split-phase 220 with a higher current capacity, allowing much higher power consumption (and cooling/dehumidifying rates).

The air coming out of your A/C unit was cooled down enough to make moisture condense on the evaporator coils; when it comes out of the A/C, it’s going to be at or very near 100% relative humidity, so yes, it’s going to feel like going outside in the early morning when there’s heavy dew on everything.

The air coming out of your dehumidifier was cooled down enough to make moisture condense on the evaporator coils, but then the waste heat from the condenser was dumped back into that air, raising its temperature and lowering its relative humidity. With lower relative humidity (due to its higher temperature), the air coming out of your humidifier will feel drier than the air coming out of your HVAC, even if it has the same absolute humidity.

And thus the net effect on the humidity of the room is the same, regardless of how the air feels when it’s blowing out of the unit. The trick with a dehumidifier is that it actually warms the room somewhat, from the heat dissipated by the compressor and the latent heat of the condensing water vapor. So they’re really only useful in spaces that have high moisture but are also naturally cool, like basements. That’s why a normal a/c unit won’t be much help down there, because it will have to run so long to dehumidify the air that it will get super cold, likely colder than the thermostat will even allow (most don’t go much below 65ºF). Then you’re fighting the fact that colder air with the same absolute amount of moisture has a higher relative humidity, so it fights against itself. If you only had an a/c and needed to dehumidify a cold room, you’d then have to also put an electric space heater down there to run at the same time, at which point you’ve basically created a dehumidifier, only it uses much more power.