June is a bit over-protective of her son, Larry. When she witnessed villain and hero fighting, she failed to comprehend what was going on. She loves her son, and is highly over-protective due to not understanding any of his interests. She only knows what she fears; that he does not live in the real world and is "not like other nineteen-year-olds!".
Whether she is on Ann's side or James' side of the family is unknown, but it's likely that she is from Anne's side given that neither are ever shown at any Possible family events and none of the Possibles are around or mentioned during "Larry's Birthday". She is seen more with James than Ann though. She may also have just married into the family, and not be related to Ann or James. If her last name is Possible, she probably does not live in Middleton.
In "Attack of the Killer Bebes", it was mentioned that there was only one Possible family living in Middleton. She may live in Upperton or Lowerton. It is also unknown if her son Larry, has ever attended Middleton High School. During "Homecoming Upset", he did go to "Meet The Queen".
June's husband is never mentioned.
Larry is the over-protected son of June. He is 19  and still lives with his parents, although he spends most of the day in a gaming and sci-fi club.
Kim Possible is one of June's nieces. Kim was often asked to look after Larry, both because of June's high-strung over-protectiveness, and because Larry clearly did need some occasional reminders of the difference between fantasy and reality.
Jim and Tim Possible are June's nephews.