Both sentences are correct.
In the first example,"I remember your talking about how your father died."
"talking"
"I remember your talking about how your father died." "talking" is a gerund
gerund or verbal noun. (The talking about the subject becomes the import portion of the sentence.)
In the second example "I remember you talking about how your father died."
"talking"
"I remember you talking about how your father died." "talking" is a present participle.
present participle. (What is remembered is the important part of the sentence.)