1. Hereditary succession via Salic law (male-only primogeniture).
That is to say succession by males to the exclusion of females. This was invoked by Philip V of France to exclude his niece Joan of Navarre.
2. Male Collaterality.
This means male succession only through the male line. This concept is also often equated with the Salic law, as it is the logical consequence of the Salic law. This was invoked by Philip VI of France to counter the claim of Edward III of England via his mother Isabella. The logic used is that a person cannot pass on a claim that one does not possess for oneself.
3. Continuity of the Crown.
With the death of the previous king, the heir immediately becomes king. This is why the pretenders of France are considered de jure kings and are given regnal names and numbers, despite not being in actual possession of the throne.
4. Inalienability (unavailability) of the Crown.
The crown is not the property of the king. In general, the king cannot appoint his successor, renounce, or abdicate.
The king must be Roman Catholic. Added by Arrêt Le Maistre of 1593, which ended the French Wars of religion. There is debate about whether Catholicism is a prerequisite to being in the line of succession or whether conversion is allowed. Considering the rigidity of the fundamental laws, I favour the former and think it is a prerequisite.
An exact date for this requirement is not entirely known, but likely emerged in the 16th century, though it is possible it emerged earlier. The aforementioned Arrêt Le Maistre of 1593 states:
JUDGMENT of the sitting parliament in Paris which annulls all treaties made or to be made which would call to the throne of France a foreign prince or princess, as contrary to the salic law and other fundamental times of the state.
Paris, June 28, 1593
The court, on the above remonstrance made to the Court by the Attorney General of the King and the subject matter of deliberation, said court, all the assembled chambers, having, as it never had other intention other than to maintain the catholic, apostolic and roman religion and the state and crown of France, under the protection of a good king very Christian, catholic and French,
Ordered and ordered that remonstrances will be made this afternoon by Master Jean Lemaistre President, assisted by a good number of advisers in the said court, to the Duke of Mayenne, Lieutenant General of the State and Crown of France, in the presence of the princes and officers of the crown, now being in this city, that no treaty be made to transfer the crown into the hands of prince or princess foreigners;
That the fundamental laws of this kingdom be kept and the judgments given by the said court for the declaration of a Catholic and French king executed; and that it is necessary to employ the authority which has been committed to him to prevent that, under pretext of religion, be transferred in foreign hands against the laws of the kingdom; and to provide as quickly as possible for rest at the relief of the people, for the extreme necessity in which it is reduced; and yet, at the present time, said court declares all treaties made and to be made hereafter for the establishment of null foreign prince and princess and of effect and value, as done to the prejudice of the Salic law and other fundamental laws of the state. [Emphasis added]