Starting with your last questions first:
The object of the verb "remember" is "part", "the part [of the presentation]", and "of the presentation" is a prepositional phrase used to modify "part" (clarifying what kind of part is meant). The second prepositional phrase in the sentence ("in the conference"), could either be considered part of the clause beginning with "people remember....", or it could be understood as a separate, additional modifier of "part". There really isn't much difference in meaning between those two interpretations.
As to your first question, about whether that or which is optional: in your sentence, "that" or "which" can be omitted before the clause "people remember the most". This is common in normal idiomatic English. Also, according to englishgrammar.org:
The relative pronoun that can be left out when it is the object in a relative clause. Here is the example they give,
This is the woman that we were talking about.
This is the woman we
were talking about.
Their example is analogous to the sentence about the mosquito moment, so we do not need to use "that" or "which".
It is true that the same website also states, as given in Zeeshan Ali's answer,
That cannot be dropped after nouns. (same source)
Since "woman" is a noun, I am confused by their apparently contradictory rules. I would go with common usage, and say that the sentence you quote is correct.