The primary function of the past perfect tense is to talk about the "past before the past". A past perfect sentence creates a chronological order and link between two events in the past. In some cases past simple tense could be used to convey the same information, but past perfect tense should be used as it makes the order of events much more unambiguous and the sentence structure more natural.
In the first example there are two events/actions:
"She sent me a new book as soon as I had finished the book."
"She sent me a new book" is an event in the past. "I had finished the book" is the past perfect portion, and signals that the book was finished prior to the new one being sent. Since the new book was sent in the past, the book was read in the "past before the past". It may also be helpful to visualize this via a timeline, an example of which can be found here.
Now apply these same rules of ordering events to the second example:
"Do you remember when my son got sick last Tuesday and I had to go pick him up? That's why I got worried when his teacher called me this morning."
We can infer that the son got sick prior to having to go to pick him up, and likewise that his teacher called prior to getting worried. So now we change it to past perfect tense to get:
"Do you remember when my son had gotten sick last Tuesday and I had to go pick him up? That's why I got worried when his teacher had called me this morning."
The third example is a little bit different because it uses reported speech (apparently incorrectly, but that is another topic).
"He said that the moment he first met her, he felt something special and began to keep a diary."
However we can still infer the order of events and change to past perfect tense to be more clear. We infer that feeling something special came prior to keeping a diary. Then the sentence changes to:
"He said that the moment he first met her, he had felt something special and began to keep a diary."
- Past perfect tense links two (or more) events which each occured in the past
- Past perfect is preferred over past simple for decriptions of two (or more) events in the past
Why should I use past perfect at all in my "Do you remember when my son got sick" example? Am I not explaining things in chronological order? I thought I was starting to grasp this past perfect thing, but now I'm even more confused.
Hopefully this is more clear now, the past perfect is used in this sentence to emphasize that there were two events, which happened in a certain order, both in the past.
Also, Why is it not "He said that the moment he had first met her, he had felt something and had begun to keep a diary"?
Didn't the meeting take place before his saying it? Or is "first met her" modifying what kind of moment it was? Also, didn't the "feeling something" and "beginning to keep a diary" happen simultaneously?
Good questions. The phrase "the moment he first met her" is specifying a time, just the same as "as soon as" and "last Tuesday" did in the first and second examples, it isn't itself an action/event. "Feeling something" and "beginning to keep a diary" do sound like they are happening simultaneously in the original example sentence, even though one is obviously the cause of the other. This further demonstrates how ambiguous using past simple instead of past perfect can be. The entire purpose of the past perfect tense is to have a clear timeline and no ambiguity of what happened first.
I apologize in advance for the wall of text. Hopefully it has enough detail to give you a clear understanding.