When do you use i.e., and when do you use e.g., and what do they mean?

 The Latin abbreviations "i.e." and "e.g." come up very frequently in writing and would probably come up more often if people were more sure of when it is right to use "i.e." and when "e.g." is required. To me, the only way to figure it out is to know what they stand for.
i.e.
"I.e." stands simply for "that is," which written out fully in Latin is 'id est'. "I.e." is used in place of "in other words," or "it/that is." It specifies or makes more clear.
e.g.
"E.g." means "for example" and comes from the Latin expression exempli gratia, "for the sake of an example," with the noun exemplum in the genitive (possessive case) to go with gratia in the ablative (prepositional case). "E.g." is used in expressions similar to "including," when you are not intending to list everything that is being discussed.

Examples of i.e. and e.g.:
1. Places to Concentrate:
I.E. (Id Est)
I'm going to the place where I work best, i.e., the coffee shop.
[There is only one place that I am claiming is best for my work. By using "i.e.", I am telling you I am about to specify it.]

E.G. (Exempli Gratia)
At the places where I work well, e.g., Starbucks, I have none of the distractions I have at home.
[There are lots of coffee shops I like, but Starbucks is the only international one, so it's the only "example" that would work.]

2. Helen and Her Siblings:
I.E. (Id Est)
The most beautiful human in Greek mythology, i.e., Leda's daughter Helen, may have had a unibrow, according to a 2009 book on Helen I'm reading.
[Helen, whose beauty launched the Trojan War, is considered the most beautiful woman from Greek mythology. There is no contender.]

E.G. (Exempli Gratia)
The children of Leda, e.g., Castor and Pollux, were born in pairs.
[The pair of boys, who are named Castor and Pollux, might be called twins, but it's not so clear about another set of Leda's offspring. Helen was said to have been hatched from an egg; Clytemnestra, born. Despite this distinction in manner of birth, Leda gestated/brooded multiple "pairs" of children, so Castor and Pollux are an example.]

Italics
I.e. and e.g. are such common Latin abbreviations that they do not require italicization.

Capitalization
If the form "I.e." looks odd, it's because both "i.e." and "e.g" are usually mid-sentence, surrounded by commas, so they are unlikely to be seen with sentence initial capitals.

Someone quoted this article to insist that i.e. should not be capitalized. The writer began his sentence with "i.e." Evidently I wasn't clear enough that rules about sentence capitalization (or other things) may take precedence over the fact that i.e. isn't generally capitalized. I was only trying to explain why "I.e" might look funny to readers.