Thinking of the words my, your, his, her, its, our, and their as adjectives seems to make a lot of sense. They precede and modify nouns, and tell "which" noun(s) is/are being referred to. In fact, the traditional grammar that many of us learned in school and that is still widely taught calls them "possessive adjectives." They seem to fill the "job description" for adjectives.
Why, then, do some modern linguistic grammars call them "possessive pronouns"?
For a couple of reasons. First, they are analogous to possessive nouns, as in
1) The president's/his automobile was surrounded by armed troops
2) I took the evidence I needed from Katrina's/her handbag
Just as president's and Katrina's are considered the genitive form of nouns, the possessives his and her, as well as the others in the set, are considered the genitive form of pronouns. Huddleston and Pullum, in The Cambridge Grammar of the English Language,* call these forms dependent (as contrasting with the "independent" forms mine, yours, his, hers, its, ours, and theirs).
Second, these words can be coordinated with possessive nouns:
3) He asked me to take care of his and Jacob's snake while they were away
Finally, according to these sources, this form acts as the subject of a nonfinite subordinate clause:
4) No one objected to my taking a third helping of the souffle
Quirk et al., in A Comprehensive Grammar of the English Language,** call these forms "weak," as contrasted with the "strong" forms mine, yours, etc. Their function is called "determinative," while the function of the "strong" versions is called "independent."
These are the arguments used by two comprehensive modern grammar authorities.
A different description is offered by Biber et al. in the Longman Grammar of Spoken and Written English*** They classify these words neither as pronouns nor as adjectives. In much the same way that they (and others) classify the demonstratives this (house), that (lesson), these (people), and those (answers) as determiners (not adjectives) when they modify nouns, they classify my, your, his, her, its, our, and their as possessive determiners. This classification avoids putting these words into part-of-speech pigeonholes into which they do not fit very comfortably.
So there is no single label that satisfies everyone. Traditional grammar sees adjectives; Quirk as well as Huddleston and Pullum see pronouns; Biber et al. see possessive determiners. Take your pick, and you won't be wrong--to someone, at least.
Marilyn Martin
*Cambridge University Press, 2002
**Longman, 1985
*** Longman, 1999