I recently came upon the sentence:
Мы хотим мира.
We want peace.
The first thing that went through my head was, This must be one of those animate masculine nouns that declines for the accusative case. Wanting to be sure, I decided to look it up on a declension table and discovered that my assumption was wrong. The word "мир" does not decline for accusative case and the role "мира" plays in this sentence is that of a noun in the genitive case. So, I started to plug and play various nouns after the phrase "I want" (not just the masculine inanimate), and have come to the conclusion that "мир" declines into the genitive because it is an uncountable noun and thereby conveys the notion of "I want (some) peace." This then also explains these constructs:
Мы хотим воздуха.
We want (some) air.
Мы хотим воды.
We want (some) water.
Мы хотим мяса.
We want (some) meat.
All of the nouns in the sentences above are in the genitive case.
After experimenting with various words, I've come to other observations about use of the genitive in lieu of the accusative:
This appears to happen most often, if not exclusively, with uncountable nouns.
Just because a genitive declension was chosen doesn't necessarily mean that an accusative declension couldn't have been chosen instead.
If you see the use of a genitive declension instead of the accusative, you can assume that some sort of notion of "some" or "some of" is being conveyed.
Favoring the genitive over the accusative is not a given for all uncountable nouns. (The word for "mustard" -- горчица -- is a good example.)
Using the genitive declension over the accusative may be more common with some words. In addition to my observation that this happens a lot with uncountable nouns, within that subset there may be tendencies to do this with some uncountable nouns but not others which may verge on the point of being something akin to a "fixed phrase." (Can any native or near fluent Russian speakers provide any examples of this?)
The gender of an uncountable noun is irrelevant. Uses of the genitive over the accusative can be seen across the genders. In this post, I inadvertently managed to provide examples of all three genders -- мир (masculine), вода (feminine), and мясо (neuter).
I didn't mean to do this when I started drafting up this question, but between what I've already read/learned and intertwining that with some personal observations, I may have answered some of my own questions. Even so, it's always best to get feedback from others when you're trying to make sense of something, so I've decided to leave this as is in the hopes that those more knowledgeable about the Russian language can support/refute/add to what I've observed and/or recommend some helpful resources on the topic.