Coach Doogan, as her name suggests, is the PE (Physical Education) coach at Hazelnut Middle School who teaches Pepper Ann and the other students and a recurring character in Pepper Ann.
In "In Support Of" she informs Pepper Ann's class that they will need "support" for the upcoming trampoline lesson. Pepper Ann misinterprets this as the coach saying that she needs a bra and spends the episode worrying and shopping for a bra. She eventually finds out what Coach Doogan meant by "support": a partner to help you on the trampoline.
Like most of the other teachers, not a lot is known about her life outside of school. In fact, the only major time we've seen her away from school was in "The Sisterhood", when she attends the "Adamant Eve" women's retreat.
She apparently used to be a nun, as she'll mention how she "left the convent" in various episodes. It it unknown when she was a nun or why she left, however.