Firstly: The Family of the Prophet ﷺ
The family of the Prophet ﷺ for whom charity is haram are the Children of Hashim  These are the descendants of `Ali, `Abbas, Ja`far, `Aqil, and al-Harith ibn `Abd al-Muttalib. only. This is the position of the majority: Hanafis, Malikis, Hanbalis, and a group of the scholars.
Secondly: The Ruling of Giving Charity to the Family of the Prophet ﷺ
It is not permissible to give charity to the family of the Prophet ﷺ,  Consensus ahs been quoted on this generally by a number of scholars, like Ibn Qudamah, al-Nawawi, and al-Qarafi. and this is even if they are not given a fifth.  Ibn Taymiyyah and Ibn `Uthaymin deemed it permissible to pay charity to them if they were not given their fifth. This is the position of the majority: Hanafis, the correct position among Shafi`is, and Hanbalis.
Thirdly: The Disbeliever
Legal charity is not given to a disbeliever.  Other than those whose hearts are to be won among them. Ibn Qudamah and Ibn al-Mundhir quoted consensus on this.
Fourthly: Giving Charity to Relatives Who Are One's Financial Burdens
It is not valid to give charity to family members who are one's financial burden to start with. Ibn al-Mundhir quoted consensus on the impermissibility of giving legal charity to one's parents, and Abu `Ubayb al-Qasim ibn Sallam quoted it as impermissible for one's children.
Fifthly: Giving Charity to Family Members Who Are Not One's Financial Burdens
It is permissible to give charity to those who are not financially dependent on the giver. This is by agreement of the four schools of jurisprudence.
Sixthly: Giving Charity to Family Members Who Are One's Financial Burdens When One is Unable to Provide
It is permissible to give legal charity to family members who are financially dependent on the giver if he is unable to normally provide for them what it is wajib upon him. This is explicitly mentioned by the Hanbali jurists, and it is the chosen position of Ibn Taymiyyah and Ibn `Uthaymin.
Seventhly: The Wife
It is not permissible to give legal charity to one's wife. Consensus has been quoted on this by: Ibn al-Mundhir, al-Kasani, and Ibn Qudamah.
Eighthly: The Wife Giving Charity to Her Husband
It is permissible for a woman to give legal charity to her husband if he is within the eight categories. This is the position of the Shafi`is, Abu Yusuf and Muhammad ibn al-Hasan from the Hanafis, a position among the Malikis, and a position among Hanbalis. It is the chosen position of Abu `Ubayd al-Qasim ibn Sallam, Ibn al-Mundhir, Ibn Hazm, Ibn Qudamah, al-Shawkani, Ibn `Uthaymin, and the verdict of the Permanent Committee.
Ninthly: The Wealthy
It is not valid to give charity to a wealthy person within the eight valid avenues of expenditure.
Tenthly: One Who Earns What Suffices Him
It is not permissible to give charity to a person who earns enough. This is the position of the Shafi`is, Hanbalis, and us the choice of Ibn Baz and Ibn `Uthaymin.
Eleventhly: The Slave
The one giving charity does not give it to his slave. Ibn Qudamah quoted consensus on this.