The Player's Handbook on the ranger's extraordinary ability favored enemy says
At 1st level, a ranger may select a type of creature from among those given on Table 3–14: Ranger Favored Enemies. Due to his extensive study on his chosen type of foe and training in the proper techniques for combating such creatures, the ranger gains a +2 bonus on Bluff, Listen, Sense Motive, Spot, and Survival checks when using these skills against creatures of this type. Likewise, he gets a +2 bonus on weapon damage rolls against such creatures. …If the ranger chooses humanoids or outsiders as a favored enemy, he must also choose an associated subtype, as indicated on the table. If a specific creature falls into more than one category of favored enemy (for instance, devils are both evil outsiders and lawful outsiders), the ranger’s bonuses do not stack; he simply uses whichever bonus is higher. See the Monster Manual for more information on types of creatures. (PH 47)
There's nothing there about immunity to critical hits or precision damage. And, after the chapter on skills, there's nothing in the Player's Handbook about the extraordinary ability favored enemy. (The Monster Manual mentions the special ability favored enemy only by listing some creatures' preferences.)
The Rules Compendium on Precision Damage says that precision damage
includes sneak attack and other abilities that work like it, such as a ninja’s sudden strike (Complete Adventurer 8) and scout’s skirmish (Complete Adventurer 12). (42)
And, seriously, the extraordinary ability favored enemy doesn't work like those special abilities. (Besides, why use an example from Complete Adventurer when there's a fine example in the Player's Handbook that can be used instead? That's just weird, right?)
However, the Dungeon Master's Guide mentions the special ability favored enemy in the context of Invisibility:
Invisibility does not, by itself, make a creature immune to critical hits, but it does make the creature immune to extra damage from being a ranger’s favored enemy and from sneak attacks. (295)
And, likewise, in the context of Darkness:
Creatures blinded by darkness lose the ability to deal extra damage due to precision (for example, a ranger’s favored enemy or a sneak attack). (302)
This leads to the question: Is the ranger's favored enemy bonus on damage precision damage or just a bonus on damage against certain creatures? That is, are creatures immune to critical hits likewise immune to the ranger's extra damage from the extraordinary ability favored enemy, or can a ranger, for example, pick as his favored enemy the creature type construct, elemental, ooze, plant, or undead as a favored enemy and against such creatures always apply that favored enemy bonus on damage to such creatures?
What I'm really trying to hammer out is the epic feat Improved Manyshot, which says
Regardless of the number of arrows you fire, you only apply precision-based damage (such as sneak attack damage or the ranger’s favored enemy bonus) once. If you score a critical hit, only one of the arrows deals critical damage (your choice); all others deal normal damage.
Emphsis mine. So addressing this feat specifically makes an answer more useful. Also, whenever I ask a question that I think only has two answers, there're inevitably more, so if you've an answer that goes unaddressed here, that's okay. Finally, while I understand the rules for primary sources and that the feat Improved Manyshot can't alter them, the conflict between the Player's Handbook and the Dungeon Master's Guide nonetheless concerns me.
Best Answer
The primary source on Favored Enemy is either Player’s Handbook in the ranger class description, or Rules Compendium if you buy its assertion of primacy. The descriptions in the Invisibility description in Dungeon Master’s Guide, Improved Manyshot in Epic Level Handbook, or even darkness, despite also being in the Player’s Handbook, are definitely not the primary source on Favored Enemy.
Further caveats, limitations, and addenda not mentioned in the primary source description are contradictions with that description. If Favored Enemy had defined itself as precision damage, the primary source on precision damage would apply, but it didn’t, which means that neither the precision damage description nor anywhere else can define it as such.
I would be inclined, in general, to follow contradictory rules as far as they go. Darkness and Invisibility cannot define Favored Enemy as precision damage in general, but it can say Favored Enemy doesn’t work in those conditions. Here, specific-trumps-general: rather than trying to redefine what Favored Enemy is (in which primacy asserts itself), they can define a special case which acceptably contradicts the general rules. So my reading of the rules as written would be that Favored Enemy is not precision damage, but it does fail to work in cases of darkness or Invisibility, and does apply only once to Improved Manyshot.
Ultimately, however, I would mostly ignore any and all rules as written that apply any more needless limitations on Favored Enemy. Of all the iconic core class features,1 Favored Enemy is one of the weakest, and that’s even assuming that it “just works” on any and all attacks against the designated foes (including, therefore, Improved Manyshot).