You are already familiar with adjectival possessive pronouns: instead of saying "the man's black hat" you may say "his black hat," etc. The main German singular possessive pronouns are ihr and sein, usually translated as "his" and "her."
In German, der Mann is a grammatically masculine noun, but in any language, a man is naturally masculine too: he is the male of the species. The same rule applies to die Frau: grammatically feminine in German, but naturally female. Thus, the possessive pronoun for der Mann is sein (his) and the possessive pronoun for die Frau is ihr (her), and it follows that the grammatical genders and natural genders are matching in these cases:
Michael Jackson hielt im Übermut sein Baby aus einem Hotelfenster.
Michael Jackson held his baby boisterously out of a hotel window.
Captions 24-25, Berlin - Hotel Adlon feiert 15 Jahre NeueröffnungPlay Caption
Astrid North hat ihr neues Album in Eigenregie produziert.
Astrid North has self-produced her new album.
Caption 28, Astrid North - Solo-DebütPlay Caption
The possessive pronoun of any grammatically masculine German noun is also masculine, just as any feminine noun takes a feminine pronoun. However, English does not translate the gender-specific sein and ihr respectively to "his" and "her" when referring to non-animate objects and many animals, but uses the gender-neutral "its" instead:
… anhand eines massiven Sterns, der sein Leben aushauchte.
… based on a massive star which exhaled its last breath.Play Caption
Die Nordsee. Sieben Länder teilen sich ihre Küsten.
The North Sea: Seven countries share its coastlines.Play Caption
As you see above, non-animate nouns follow their gender equivalents in German, but in English both are translated as "its" rather than "his" or "her." But how does German deal with pronouns of neuter nouns?
Für jeden von uns bringt das neue Jahr seine ganz eigenen Herausforderungen mit sich.
For each one of us, the New Year brings its own very [unique] challenges with it.
Caption 2, Angela Merkel - NeujahrsansprachePlay Caption
Im Wechsel der Gezeiten verändert das Watt ständig sein Gesicht.
With changing tides, the intertidal zone is continuously altering its appearance.Play Caption
As you see in the examples above, German does not have a unique possessive pronoun for neuter nouns, but instead reverts to the masculine pronoun sein. This can lead to the rather odd situation of a grammatically neuter pronoun such as das Mädchen—which is naturally feminine in that "the girl" is a female of the species—taking the masculine pronoun sein:
Schnell nahm das Mädchen ein neues Streichholz in seine zitternden Hände.
The girl quickly took a new match in her trembling hands.Play Caption
As you see, seine was translated to "her"—not "his"—in this case, because only "her" is correct in English. Although it is formally correct to use sein for das Mädchen, it is very common to see feminine pronouns such as ihr used for grammatically neuter but naturally feminine figures. In fact, in the video quoted above, every single pronoun reference to das Mädchen actually used grammatically incorrect feminine pronouns! So while it is good to be conscious of this rule, it won't sound unnatural if the occasional Mädchen winds up as ihr in your German!