Rules of Inheritance in Islam
Question

What are the most important rulings on shares of inheritance?
Praise be to Allah.Importance of knowledge of Islamic law of inheritance
Knowledge of Al-fara’id (allocated shares of inheritance) is one of the most important branches of Islamic knowledge. In three verses of Surah An-Nisa’, Allah, may He be Exalted, states many of the rulings thereon, then the Prophet’s Sunnah explains these rulings in detail.
The Companions (may Allah be pleased with them) paid attention to the knowledge of shares of inheritance, as did the Tabi`in (the generation who came after the Companions) and the scholars who came after them, and many books were written about this branch of knowledge.
We will mention below some of the main guidelines on this branch of knowledge:
Components of inheritance in Islam
The essential components of inheritance are: 

the heir, 
the testator (that is, the deceased),
and what is to be inherited (the estate).

Conditions of inheritance in Islam
The conditions of inheritance are three:

Ascertaining that the heir is alive at the time of the testator’s death, or that he is to be regarded as being effectively among the living, such as a foetus in utero. The foetus may inherit subject to conditions: Firstly establishing that he was in the womb at the time of the testator’s death, even as a  Nutfah (fertilised egg); and, secondly, establishing that he was born alive and in stable condition.
Ascertaining that the testator has died, or that he is to be regarded as being effectively among the dead, such as one who is missing.
Knowing who may inherit, the reason why he is entitled to inherit, his connection to the testator and how close that connection is, and so on.

Causes of entitlement to inheritance
The causes of entitlement to inheritance are three: 

Nikah (meaning a valid marriage contract only; it is not stipulated that the marriage should have been consummated); 
Wala’ (manumission of a slave); 
And Nasab (blood ties).

Impediments to inheritance

The impediments to inheritance are three: 


enslavement (a slave cannot inherit anything); 
murder (the killer cannot inherit anything from the one whom he killed); 
and difference of religion (a disbeliever cannot inherit from a Muslim and a Muslim cannot inherit from a disbeliever).

Categories of heirs among men
There are fifteen categories of heirs among men: the son; the son’s son , no matter how far the line of descent reaches; the father; the paternal grandfather, no matter how far the line of ascent reaches through the male line only; the full brother ; the half-brother through the father; the half-brother through the mother; the son of a full brother; the son of a half-brother through the father, no matter how far the line of descent reaches; the paternal “full-uncle” (full brother of the father); the paternal “half-uncle” (the half-brother of the father through their father), no matter how far the line of ascent reaches; the son of the paternal “full-uncle” (full brother of the father), the son of the paternal “half-uncle” (the half-brother of the father through their father) no matter how far the line of descent reaches; the husband ; and the freed male slave.
Categories of heirs among women
There are ten categories of heirs among women: the daughter ; the son’s daughter, no matter how far her father’s line of descent reaches; the mother ; the maternal grandmother; the paternal grandmother; the full sister; the half-sister through the father; the half-sister through the mother; the wife; and the freed female slave.
Types of inheritance in Islam
Inheritance is of two types: Fard (pl. Fara’id – allocated shares) and Ta`sib (residual inheritance).

Fard refers to the shares that are allocated to heirs in accordance with the religious texts, such as one half, one quarter, one third, and so on.
Ta`sib refers to what an heir may receive from what is left (the residue) of the estate after those who are entitled to allocated shares have received their shares.

What are the allocated shares of inheritance?
The allocated shares that are mentioned in the Book of Allah, may He be Exalted, are six: one half, one quarter, one eighth, two thirds, one third and one sixth.
First of all, those who are entitled to allocated shares take their shares. Then if anything is left of the estate, it is taken by the residual heirs. If there is nothing left for them, then they have no share, in accordance with the words of the Prophet (blessings and peace of Allah be upon him): “Give the shares of inheritance to those who are entitled to them, and whatever is left goes to the closest male relatives.” (Narrated by Al-Bukhari, 6732 and Muslim, 1615)
Books on inheritance on Islam
As for the details on the rulings of inheritance, and discussing the case of each heir and the conditions of his inheritance, in all circumstances, that requires detailed discussion for which there is no room here. Please refer to the books that have been written on this topic, the easiest of which include:

Al-Fawa’id Al-Jaliyyah fi’l-Mabaahith Al-Fardiyyah by Shaykh `Abd Al-`Aziz ibn Baz (may Allah have mercy on him)
Tas-hil Al-Fara’id by  Shaykh Muhammad ibn `Uthaymin (may Allah have mercy on him)
At-Tahqiqat Al-Mardiyyah fi’l-Mabahith Al-Fardiyyah by Shaykh Salih Al-Fawzan (may Allah preserve him).

And Allah knows best.