Q:

What exactly is "base AC"?

The phrase "base AC" appears in the D&D 5E basic rules exactly once, in the description of the Mage Armor spell.  This says:

The target’s base AC becomes 13 + its
  Dexterity modifier.

But if we expand the acronym AC, "Base Armor Class" also appears: but again, just once, as far as I can see. This is in the equipment chapter, and armor and shields, and says:

The armor (and shield) you wear determines your base Armor Class

By this, it seems logical to assume that "base Armor Class" is the armor value, possibly plus 2 for a shield. But, in the case of Mage Armor, dexterity is specifically included in the "base AC". Is it intended that dexterity is added on top of that, for a total AC of 13 + dex modifier + dex modifier again?
If you look at the details for light, medium, and heavy armor, each of these specifies individually how Dex is treated (always added to Light, added up to +2 for Medium, and never factored in even if negative for Heavy). That sounds like the influence of Dex, if any, is included in the part of the armor, and therefore in your base Armor Class. But, on the other hand, these descriptions refer to "the base number from your armor type" — so is there a "base number" that's different from your "base AC"?
The question How does barkskin work? exemplifies how this confusion (for those of us used to more complicated AC from previous edition) trickles down: the question asks if something affects "total AC, Base AC or just armor AC". So my question, fundamentally, is: is "base AC" actually a special term in 5E, with a coherent meaning, or is it just some words that get used sometimes in describing armor class sometimes? If it is a definite term, what exactly does it include?

A:

Your "base AC" is your AC before any modifiers. However, it is not a defined game term at this time as it has little or no use outside of the Mage armor spell.
Such modifiers may include:

Class based bonuses like from a Fighting style
Magic bonuses from items
Temporary bonuses from spells.
Other things not included in this list.

So your "Base AC" is either your armor plus your ability (if applicable) plus your shield, or your natural armor plus your Dex or other stats as applicable. Also Mage Armor would set your "base" AC to 13+dex.
Your Base AC includes any applicable stat bonuses. So "base" in the case of Mage Armor is 13 + Dex, there is no additional bonus to that. A shield modifies your Base AC, and increase it by 2.
The armor + shield also includes your Dex for medium and light armors. So your Base AC in the following armor types is as follows:

None: 10 + Dex
Leather: 11 + Dex
Hide: 12 + Dex (max 2)
Chain: 16
Chain + Shield: 18
Mage Armor: 13 + Dex

Various other things set your base AC. But it's based on both your AC and the appropriate stat.
Lastly, there is no intent for there to be a difference between Unarmored Defenses and Mage Armor.

Wax Eagle: Is there a reason why the wording is inconsistent between Mage Armor and Unarmored Defenses? Mage Armor uses "Base AC"
Jeremy Crawford: The difference isn't intentional.

A:

Your base AC is exactly what your current AC calculation method says. Base AC is not a defined term, but to me it represents the less situation part of your AC. Current/total AC = base AC + temp/situational bonus/penalties (like those given from the shield spell).
You will not be adding your dex mod twice when using Mage Armor.
You only (sometimes) add your DEX to your AC because the calculation method for AC while wearing armor explicitly tells you to.
From Basic Rules pg 9, Under the Armor Class:

Without armor or a shield, your character's AC equals 10+ his or her
  Dexterity modifier. If your character wears armor, caries a shield, or
  both, calculate your AC using the rules in chapter 5. 
...
Some spells and class features give you a different way to calculate
  your AC. 
  ...

pg 43 chapter 5 - equipment

Armor Class (AC): Armor protects its wearer from attacks, The
  armor(and shield) you wear determines your base Armor Class

Each type of armor (light, medium, heavy) then has a section that explains if/how your Dex mod is added.
From the Light Armor section on pg 43:

If you wear light armor, you add your Dexterity modifier to the base
  number from your armor type to determine your Armor Class.

