I hope you don't mind my staying with you tonight.

[B45]

I hope you don't mind my staying with you tonight.

VS

I hope you don't mind me staying with you tonight.

Are both okay?

 

[Rover_KE]

Yes.

 

[MikeNewYork]

Yes. In most cases possessive adjective + gerund is equivalent to objective pronoun + present participle.

 

[B45]

Yes. In most cases possessive adjective + gerund is equivalent to objective pronoun + present participle.

I don't understand the texts in bold.

 

[MikeNewYork]

Those are the names of the uses you brought up. If you don't understand them, I suggest you Google them and read some web pages.

possessive adjective
gerund
objective pronoun
present participle

 

[Rover_KE]

Alternatively, just please yourself which you use — they are both correct.