What are the differences between these two abbreviations? What are the appropriate situations to use each?
As reported by the New Oxford American Dictionary:
e.g. /ˌi ˈdʒi/
ORIGIN from Latin exempli gratia (for the sake of an example).
i.e. /ˌaɪ ˈi/
that is to say (used to add explanatory information or to state something in different words): a walking boot that is synthetic, i.e., not leather or suede.
ORIGIN from Latin id est "that is."
E.g. is used in expressions similar to including, when you are not intending to list everything that is being discussed.
I.e. is used when what you are introducing is equivalent to or an explanation of what comes before it in the sentence.
I like root vegetables, e.g., potatoes.
I like root vegetables; i.e., the ones that grow underground.
There's a neat way to remember which is which: "e.g." and "example" both start with the letter "e" (so read "e.g." as "for example"); and "i.e." and "in other words" both start with the letter "i".
Please see the most authoritative source: The Oatmeal.