"I'm sorry I gave you such a miserable time, Harry, but the Dursleys would never have let you come if they'd thought you enjoyed it"
-- Arabella Figg (OP2)
"Did you expect him to let you wander around on your own after what happened in June? Good Lord, boy, they told me you were intelligent... right... get inside and stay there"
-- Arabella Figg (OP2)
Arabella Doreen Figg is a Squib, living in a house on Wisteria Walk in Little Whinging, Surrey. Mrs. Figg is an original member of The Order of the Phoenix and for fifteen years was assigned the task of watching over Harry Potter, who was growing up two streets away in Privet Drive (GF36, OP1, OP2). Harry stayed at Mrs Figg’s house every year on Dudley’s birthday, where he was forced to look at pictures of every cat she’d ever owned. He hated it at her house, but this was deliberate on her part. She saw to it that he didn’t enjoy himself too much, fearing that if he seemed to like being there, the Dursleys wouldn’t send him to her anymore (OP2).
Mrs. Figg is an eccentric woman who wears tartan carpet slippers and a housecoat when out shopping for tinned cat food at the corner shop. She wore the slippers in court as well when she testified at the hearing when Harry was accused of underage magic. Although she appears to talk to herself, other members of the Order are often present under invisibility cloaks, and her cats understand human speech (OP1, OP2) probably because they are part Kneazle (JKR). Arabella, in what seems to be typical of Squibs, owns several crossbred Kneazle cats which act as more than pets. She uses them as watchers, keeping an eye on things in Privet Drive.
Even though Harry learned about the Wizarding World in 1991 when he turned eleven, he didn’t realize until the summer of 1995 that Mrs. Figg, his old babysitter, was actually part of that world. One warm evening in August, Mrs. Figg had assigned Mr. Tibbles, her cat, to watch Harry. The cat noted that Mundungus Fletcher had abandoned his post guarding the boy and ran to tell Mrs. Figg. In the meantime, two Dementors attacked Harry and Dudley. Mrs Figg turned up after the attack and hustled Harry back to the Dursleys. Along the way, she informed Harry about her part in watching over him all those years and ordered him to stay indoors and wait for instructions.
A few weeks later, when Harry was appearing before the Wizengamot to account for his using magic during the Dementor attack, Arabella Figg testified that as a Squib she could see Dementors although Muggles cannot (OP8).
Other canon notes and references
Mrs. Figg's house smells like cabbage (PS2).
As Mrs. Figg was crossing Privet Drive on her crutches, Dudley knocked her down with his racing bike (PS3).
Mrs. Figg is a dealer of Kneazle and cat cross-breed creatures (JKR)
JKR revealed on her website that Mrs. Figg did not see the Dementors, but rather she lied at the Ministry of Magic's hearing. From this we can assume that all Squibs are unable to see Dementors, but they still are able feel the affects of them, like Muggles. Mrs. Figg is described as speaking at the hearing "as though she had learned what she was saying by heart." We can now assume she was reciting what she was told by Dumbledore. Harry thought "it sounded to him as though the most she had ever seen was a picture of a Dementor." Madam Amelia Bones says during the hearing that Mrs. Figg describes the "effect" of the attack accurately, but does not acknowledge that her physical description leaves something to be desired. It leaves one to wonder whether Amelia Bones was aware that Squibs are unable to see Dementors. She says, "...I can't imagine why she would say they were there if they weren't." Madam Bones never directly says she believes that Mrs. Figg witnesses the attack, only that she doesn't believe Mrs. Figg would lie about the events.
If Mrs. Figg was found to be lying at the hearing, we can assume that she would be disciplined or even sent to Azkaban for perjury. Her willingness to lie for Harry and Dumbledore shows her bravery and her loyalty to the Order.