Confessional poetry is a category of poetry which is, in the broadest sense, exactly what the name implies. This genre of poetry shares certain characteristics.
First it is autobiographical, which means it is generally written from a first-person point of view. What this does, of course, is add drama and impact to whateverrevelation is about to unfold. because it's real. Who isn't intrigued by hearing someone's confession of the soul?
Second, it is revelatory. In other words, I could write an autobiographical poem about getting a new car, but it may not reveal anything about who am or what I'm thinking/feeling in any way. Confessional poetry is, by its very nature, a purging of some kind of inner torment or issue or shame.
Third, it deals with traditionally "taboo" subjects, things people would not ordinarily "confess" to perfect strangers. In today's world, frankly, there are very few taboos left which are not openly discussed or written about openly; however, at the peak of the confessional poetry movement last century, there were things which were virtually never spoken. Suicide, abortion, mental disorders, abuse--these were all silent problems or tragedies known only only to the sufferers themselves.
Finally, though there are certainly no poetic "rules" regarding confessional poetry, the form is often free verse. This is not particularly surprising, as these are issue which are overflowing from the hearts and lives of the poets and generally would not suit a more structured format.
The authors of such poetry are often romanticized, but many of them do end their lives in some tragic way. Sylvia Plath, Theodore Roethke, Anne Sexton, and Robert Lowell are among the most recognized confessional poets.
I've included a good overview eNotes site featuring Sylvia Plath, below.