Do you mean a book that some parishes have that people write into? (I am clarifying because in my experience this book is a new or local phenomenon)
Generally, they are for prayer requests. They give people another place to put their concerns. Some churches allow congregants to voice their petitions during the Mass. If there are many in attendance, this can become difficult to manage. There is no requirement to use the petition book.
As for what goes into the book: Life changes, general health concerns, intentions for others or self are appropriate. These books tend to be available for anyone to read so you might want to be careful what you write, though you could be as specific as prudence dictates (if it belongs in the confessional or is gossipy, then “John is going through a difficult time” is plenty enough ).
Even though the petition book may be mentioned during the Prayer of the Faithful, it is not the same as having a Mass said for your intention. It also is not the proper way to handle urgent needs (such as requesting Sacraments - one who is in need of the Sacraments should call the parish office or priest’s direct line).
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