A public (paper) petition is a petition to the House of Commons presented by an MP. They must ask clearly for the House of Commons to take some action.
E-petitions enable members of the public to petition the House of Commons and press for action from the Government.
Groups or organisations who are affected by a Private Bill can petition against the Bill as it is considered by each House.
A member of the House of Lords may also present a petition to the House of Lords reading out who the petition is from and what action the petitioner(s) wish the House to take.