The family on the grandmother's side is part of a satanic cult (hence the pentagram on the wall behind the picture). The grandmother's mother (Katie and Kristi's great-grandmother) made a deal with a demon for wealth (this is referred to in PA2 when the girl is researching demonology). She had access to the demon since she was a satanist. In PA2, it mentions that if someone makes this kind of deal, the first-born male of the family must be sacrificed. This is why in PA3, the grandmother tries talking her daughter into having a son, because she knows that a son must be sacrificed.
The demon's name is Toby. Toby is always present with the family because he is waiting to take the soul of the first-born male. Since children are more susceptible to having a spirit or entity contact them, Toby contacts Kristi, and they become "friends". Since Toby is a demon, and therefore evil, he threatens Kristi and likes to be in control. This is why Kristi tells her stepdad that Toby doesn't like to be touched. Toby is very persistent about taking the first-born male, and wants Kristi to join the cult. So, he tells her that she must go to her grandmother's house in order to participate in the ceremony to become part of the cult. This is shown in the scene where Toby is messing with Katie, and Kristi is seen talking to Toby, and she says "Okay, I'll do it. Just leave Katie alone.". It is assumed that Toby is telling Kristi to go to the grandmother's house. This is why immediately after, Kristi goes to her mom and tells her they must go.
Once they are at the grandmother's house, the grandmother plans to take Katie and Kristi and make them part of the cult. So, Toby and the grandmother kill the parents and in a way possess Kristi and Katie so that they will join. This is why the other members of the cult are present at the house (the group of old ladies). They are going to do the ceremony to initiate the girls into the cult.
Now, it is assumed or implied that the grandmother becomes the girls' guardian, and also brainwashes them so that they don't remember what happened. In PA2, Katie and Kristi both say how they don't remember their childhood. The demon is still present when they are adults, but as explained in PA1 by the ghost expert, demons can lay dormant for years. Toby was dormant until Hunter (the first-born male) was born. He came back to retrieve Hunter's soul. He possess Kristi so that he can take Hunter, but Kristi's husband and his daughter find out that she is possessed, and transfers the demon to her sister Katie (it must be transferred to a blood relative for someone to become unpossessed by a demon, as explained in PA2). In PA1, the Toby is now following Katie, and ends up possessing her. She kills Micah, and then, in PA2, is seen killing her sister and her husband, so that she can take Hunter, which she does. This supposedly concludes the story, since Hunter is now taken by Katie/Toby for sacrifice, which would fulfill the great-grandmother's side of the deal she made with Toby for wealth.
Seen all 3 movies.