The quick answer: In most contexts, these two words are effectively synonyms, especially when discussing possibility, e.g. "(Can/Could) you [perform some action]?".
Longer answer:
There is a somewhat subtle difference in these two phrases that becomes a lot less subtle in different contexts. Let's take the words one at a time:
Can deals with actual, literal ability to perform an action. When you ask if someone can explain something, you are literally asking whether they are capable of explaining it. Edit: It is important to note that, as @J.R. has pointed out, this is by no means the only use (or even the most common use) of the word can or the phrase can you .... This construction is often abused to mean effectively the same thing as could you..., i.e. would you be willing to..., etc.
Could deals with the possibility of performing an action. It is definitely a subtle difference, but basically, when you discuss possibility, you generally are aware of their ability to do it but are tentative about their willingness or availability to perform the action. This is by no means an exact definition.
It is also worth noting that the word could is actually the past tense of can. In English, using the past tense form of a word in this way is often meant to indicate tentativeness.
I wanted to talk with you.
I wondered if you would be available this afternoon.
I thought it might a good idea.
In sentence 1, that acts as a relativizer (relative pronoun). It may be dropped (in any register) because it acts as the direct object of the verb in the relative clause. It could not be dropped in formal English (though it often is in informal spoken English) if it were the subject of the verb in the relative clause:
✲ It's the same girl Ø took our family photo.
In the remaining sentences, that acts as a subordinator (subordinating conjunction). In sentences 2, 3 and 4, that may be dropped because the subordinate clause which it heads is the direct object of the verb in the main clause and is in its ordinary position immediately after that verb.
If that played another role, such as subject, or if the subordinate clause were displaced to another position, that could not be dropped, because it would not be clear that it is in fact a subordinate clause:
✲ Ø he's protected by his family is understood by Alex ... The clause falls at the beginning of the sentence, before the verb is, because it has become the subject. That cannot be dropped.
✲ I came to know some eight or ten days after I got the report Ø you got stuck in traffic. ... Here the subordinate clause has been separated from its governing verb by a fairly long ('heavy') adverbial phrase . You could probably get away with dropping the that in speech, but it cannot be dropped in formal writing.
In sentence 5, and in these rewrites of sentences 4 and 5, the situation is a little different: These subordinate clauses are predicative complements of BE, and in speech that may be dropped even if the clause is moved to the front. In writing it's permitted, but not advisable; you really want to give the reader as many clues to your structure as possible:
? What many people are saying is Ø they saw a ghost.
? Ø I need help from you is the reason I'm helping you.
In other uses, as a demonstrative adjective or a demonstrative pronoun, that may not be dropped.
I want that puppy. but not ✲ I want puppy.
John took that from Shakespeare. but not ✲ John took from Shakespeare.
✲ marks an utterance as unacceptable
? marks an utterance as possibly unacceptable
Ø marks the place where that is omitted
Best Answer
"Whether" is nearly always replaceable by "if", for embedded questions. So
and
have exactly the same meaning: I don't know if it is true or false that Peter will come to school.
Your other sentence,
is grammatical, but in principle has a different meaning (though that meaning is unlikely with a first person subject).
If we switch to a third person subject:
means that she doesn't know the truth of the question. But
implies that it is a fact (known to the speaker) that Peter will come to school, but that Jane doesn't know that fact.
Clearly, this distinction is unlikely for a first person subject, because the subject is the speaker. For this reason, I think, you do hear
with the same meaning as if/whether; but I would say this is very colloquial.