What's the right of primogeniture?

It depends. It varies from culture to culture and even from situation to situation. Generally speaking, primogeniture refers to making the firstborn (often the firstborn son, thus bypassing any daughters born before that son, though not necessarily) the sole heir to everything owned by his or her parents. The firstborn inherits all titles, lands, property, and money, it then becoming the firstborn's. If the firstborn has died, then primogeniture gives way and the eldest surviving child (again, often the eldest son, though not necessarily) becomes the heir apparent. 

In the UK, for example, primogeniture in the monarchy meant passing the Crown and all the Crown's lands, property, and wealth to the firstborn son, but if no firstborn son, then to the firstborn daughter. So, when King George, who had no sons, died, primogeniture meant his firstborn daughter, Elizabeth, inherited the Crown and all the Crown's lands, property, and wealth, she becoming Queen Regnant, her then husband, Prince Philip, becoming not Prince Regnant but merely Prince Consort. Prince Charles is their firstborn, so he has the right of primogeniture. He gained his princedom and will, upon his mother's death, inherit the Crown from her and become King of the United Kingdom and inheriting all the Crown's lands, property, and wealth, inherit her extremely vast wealth. However, his father, Prince Philip, also has assets of his own, he being born a prince in his own right, Prince of Greece, so when Prince Philip dies, Prince Charles will inherit from his father as primogeniture, as well. Having the right of primogeniture, Prince Charles will inherit from both parents and their titles, lands, property, and wealth will all combine in him with his own wealth to become what his firstborn, Prince William, will inherit through right of primogeniture. Now Prince William's mother, Princess Diana, has already died. When she died, he would've had the right of primogeniture and inherited her substantial wealth, but since she was no longer titled nor had any inherited title other than the honorific of "Lady" via her father the 8th Earl Spencer, she was able to supersede his right of primogeniture to inherit all her wealth by making a last will and testament that superseded it and divided her wealth equally between him and his younger brother, Prince Harry, her stated reason being that she wanted both of her sons to have wealth and means outside of that provided by the Crown, by the Royal Family, so that they would not be bound to the Crown and so that if ever they so chose, they would have the means to walk away from the Crown and live independently from the Royal Family and not have that choice mean a life of destitution and poverty, which turned out to be quite prescient of Princess Diana, for her son Prince Harry has done exactly that, walked away from the Crown, from the Royal Family, and had she not done that, had she not overridden primogeniture and made him an equal heir to her wealth, he very likely could not have made the choice to do that as he could not have been able to afford that choice, afford to live independently of the Crown, especially given his lifelong need to pay for a protection detail for himself and his family and how his global fame and need for an entourage of personal protection very much limits his ability to go into business or get any kind of paying job.  

The UK recently changed primogeniture, though. So one day when Prince George, son of Prince William, grandson of Prince Charles, great grandson of Queen Elizabeth, is King, should his firstborn be a daughter and his secondborn be a son, the Crown will not pass to his firstborn son, like has been how primogeniture has always been defined in the UK's monarchy, but will instead pass to his firstborn, so if his firstborn is a daughter, primogeniture will make her his sole heir. 

From as far back as patriarchal times, God’s servants accorded special rights to a man’s firstborn son. Upon the death of the father, his eldest son assumed the responsibilities of family head. He would care for the family and exercise authority over those members of his household who continued to dwell there. The firstborn also represented the family before God. While all sons received an inheritance, the firstborn received the principal inheritance. Compared to the amount of property the other sons might inherit, the firstborn received a double portion.

In the days of the patriarchs, the eldest son could forfeit his right as firstborn. Esau, for example, sold the birthright to his younger brother. (Genesis 25:30-34) Jacob transferred the birthright from his firstborn son, Reuben, to Joseph. Reuben lost the privilege because of his immoral conduct. (1 Chronicles 5:1) However, under the Mosaic Law, a man with more than one wife could not transfer the benefits of the birthright from the first son of one wife to the first son of another wife just because the latter was particularly beloved. The father was to respect the right that naturally belonged to his firstborn.​—Deuteronomy 21:15-17.

It's not always the case. And I even suspect that the right of primogeniture is what Satan expects, he didn't think it was fair that angels would end up serving God's latter creation; Mankind. I think that is what set the rebellion in heaven, some of the angels didn't like the idea of God creating Man for the greater birthright. The second born, if you will. So God repeated that pattern in various people in the Bible. Like King David, he wasn't the firstborn son in his generation. And Joseph, Solomon, and Jacob. None of them were the firstborn sons. Yet they held a position over all their elder brothers. The right of primogenitor went to the firstborn through the father, though God made sure some of those with the greater promise than their elder brothers. 

The oldest son in a family had a right to a double share of his deceased father's estate.  Along with it came leadership of the family and the responsibility to care for his unmarried sisters and his father's widows.

The right of succession belonging to the firstborn child, especially the feudal rule by which the whole real estate of an intestate passed to the eldest son.