the words for "his", "her" and "their" *look* the same, but there is a difference in the mutations they cause on the following word:
a chat = his cat
a cat = her cat
a gcat = their cat
a athair = his father
a hathair = her father
a n-athair = their father
As you can see, "a" lenites a following consonant when it means "he", it does nothing to consonants, but prefixes an "h" to vowels when it means "her", and it eclipses when it means "their".
The other possessive pronouns are:
mo (+lenition) = my, becomes m' before vowels or fh
do (+lenition) = your (singular), becomes d' before vowels or fh
ár (+ eclipsis) = our
bhur (+eclipsis) = your (plural)