This is my mother, Susie.

Is a comma needed after 'mother'?

Thanks.

Leave it out and you're suggesting that you have more than one mother, not an impossibility these days. Put it in and you're ambiguous as to whether your mother's name is Susie or you are introducing your mother to Susie. The latter possibility would, presumably, be cleared up by context.

The noun Susie is a nonrestrictive appositive further identifying the noun mother, and Maud should be preceded by a comma.

Who's Maud?

Cipher...
Mod squad
Harold and Maude
Or, my personal favorite
Maud Lynn
Thanks, AWAD. I cracked Espen within seconds.
Cheerio

Who's Espen?

Sounds like ESPN, sport's fans.
From the movie Blended.
Can we get a fist bump, 'Larry'?