This is not a complete explanation, but simply a reminder list. For "kinds of tone rows" and "compositional usages of tone rows", try to think in a style-specific way: which composers favored which types and usages of rows, and why?
Five important general considerations:
ordered, not ordered examine Po find the row forms how are the row forms used? why these row forms?
The basics:
P, I, R, RI transposition (transposition or index number) matrix order numbers (0-11, not 1-12)
Kinds of rows:
all-interval row derived row symmetrical row embedded subsets tonal implications in a row source sets
Compositional usages of tone rows:
aggregates and combinatoriality all- and semi-combinatoriality invariant subsets overlap of row forms rotation secondary set "total" serialism