I don't think there is an official set of names, and worse, there are pieces that have conflicting meanings. A "Book" is used as the whole story, as a volume or as a part. A "Part" is either A book in the series or A subsection within book.
Let me try to put it in order but take into account this is by no means ultimate or a law.
- Series/Franchise/World/Saga. There is this world/universe/multiverse where stories take place. It's self-contained and doesn't bind to any other books. It's not even limited to one author. Usually name is supplemented with the "Series/Saga/World of..."
- Storyline/Series [see, series within series?]. Following adventures of one group of characters in a single world. It's often quite non-straightforward; see an example...; usually named "character's Storyline"
- Part [of Series]/Book. A self-contained story either released alone or as a part of series. The primary unit, has its own title.
- Volume - a physical subdivision of a Book, a piece contained within one binder (or one physical file.) It frequently coincides with the Part of the Book. Usually has a number, rarely a title.
- Part [of a book]. It's a major subdivision often coinciding with volume, but not always. For example, Lord of the Rings is three volumes but six parts plus appendices. It's a major turn of events, resetting the mood and pace and often with its own climax. It may or may not have a title.
- Chapter - I think we had discussions of what comprises a chapter here. I think it's easiest to understand but hardest to put in words.
- Scene/Section - a piece delimited by
* * * or the likes, usually with a continuity of time or location.
Note there is no agreement to naming especially within the Chapter/Volume/Part area: I've seen any of them named "Book" ("Sir Thaddeus" is physically one book, one volume but it's A Nobleman's Tale from the Years of 1811 and 1812 in Twelve Books of Verse, which are actually chapters. Six parts of Lord of the Rings are called "Books".) And it's very frequent to make Volumes and Parts coincide (and then often Parts are called Volumes) but the Part-Volume relation may be entirely uneven, a part split between volumes or vice versa.