Red's mother is only mentioned briefly in the series by Rusty as he is trying to convince his mom and his dad to let him become a trainer in the episode The Journey Begins. However, Rusty's parent do not know who Red or his mother is.
While she never actually appears in person in the series, it can be assumed that she retains her sprite from the games.
She is the mother of Red. She let her son drop out of school and embark on his Pokemon journey.