What is family in Islam and what is its importance?
The family, according to the Islamic concepts, is not a mere means of satisfying animalistic instincts, nor is a family's home merely an abode for sleep; but the family in Islam is built upon a holy bond and a mutual contract of tranquility, love and kindness between two hearts.The holy Quran says:
And of his signs is that he created for you wives from among yourselves that you might reside with them, and has put kindness and mercy between you. Surely, there are signs in this for those who think. 
Muhammad (s) has said:
He who marries preserves one half of his religious commandments, so he must fear Allah regarding the other half.
In fact the family unit is an important component of Islam, and all elements of a family are given due significance – from parents to children to spouses to kith and kin. There are a lot of verses about each of the elements; but I discussed some of them that they are about parents especially mother. The Holy Quran repeatedly reminds its readers of the duties children have toward parents, particularly in their old age. God says in the holy Quran:
your lord has ordered you to worship none except him, and to be good to your parents. If either or both of them attain old age with you, do not say: "fie on you", nor rebuke them, but speak to them with words of respect. And lower to them the wing of humbleness out of mercy and say: 'my lord, be merciful to them, as they raised me since I was little.
The family, according to the Islamic concepts, is not a mere means of satisfying animalistic instincts, nor is a family's home merely an abode for sleep; but the family in Islam is built upon a holy bond and a mutual contract of tranquility, love and kindness between two hearts.
While we should obviously respect both our parents, our mothers are given special treatment in Islam. This is because our mothers carried us for nine months, and were our primary caregivers when we were children. In one saying of the Prophet Muhammad (PBUH), a companion asked him who deserves the most good treatment and respect. The Prophet (PBUH) replied, “Your mother.” The companion asked again, and again the reply was “Your mother.” The companion asked third times, and once more the Prophet (PBUH) said, “Your mother.” When the companion asked for the last time, the Prophet (PBUH) then answered, "Your father.”
In one saying of the Prophet Muhammad (s), a companion asked him who deserves the most good treatment and respect. The Prophet (s) replied, “Your mother.” The companion asked again, and again the reply was “Your mother.” The companion asked third times, and once more the Prophet (s) said, “Your mother.” When the companion asked for the last time, the Prophet (s) then answered, "Your father.”
Role and Functions of family in Islam
There are a lot of role and functions for family in Islam. I mention some of them briefly:
1) Protection and sustaining of the human race:
The Quran says:
Allah has given you wives from among yourselves, and has given you and your wives, sons and grandsons. 
2) Freedom and intactness of society from immorality:
Muhammad (s) says:
O young people, those of you who can marry let them do so for this causes abstinence and continence but those who cannot should go on fasting, which makes them refrain and it safeguards them. 
3) Keeping the community secure against disease:
As you know these days there are sorts of many infectious diseases such as AIDS and other diseases. The people can be maintained just by having the Marriage. 
4) Spiritual and Psychological ease and tranquility:
The family is the origin out of which all love, kindness and intimate familiarity springs between the spouses, their children and other family members. Any civilization that is expected to survive must be built upon love, balance and bounty. 
5) The family is a cooperative institution:
The two spouses collaborate and share responsibilities, each of them in his or her own special capacity and sphere. Consequently, children are imbued with cooperative trends and altruistic intentions. Islam splendidly urges the building of families and marriage. Muhammad (PBUH) says:
Marry and take wives for these increases your means of living. 
 Chapter 30: AL-ROOM, verse 21. وَمِنْ آيَاتِهِ أَنْ خَلَقَ لَكُم مِّنْ أَنفُسِكُمْ أَزْوَاجًا لِّتَسْكُنُوا إِلَيْهَا وَجَعَلَ بَيْنَكُم مَّوَدَّةً وَرَحْمَةً إِنَّ فِي ذَلِكَ لَآيَاتٍ لِّقَوْمٍ يَتَفَكَّرُونَ
 Chapter 17: AL-ISRA (ISRA'), verse 23 – 24. وَقَضَى رَبُّكَ أَلاَّ تَعْبُدُواْ إِلاَّ إِيَّاهُ وَبِالْوَالِدَيْنِ إِحْسَانًا إِمَّا يَبْلُغَنَّ عِندَكَ الْكِبَرَ أَحَدُهُمَا أَوْ كِلاَهُمَا فَلاَ تَقُل لَّهُمَآ أُفٍّ وَلاَ تَنْهَرْهُمَا وَقُل لَّهُمَا قَوْلًا كَرِيمًا ﴿23) وَاخْفِضْ لَهُمَا جَنَاحَ الذُّلِّ مِنَ الرَّحْمَةِ وَقُل رَّبِّ ارْحَمْهُمَا كَمَا رَبَّيَانِي صَغِيرًا ﴿24
 Bukhari, Volume 4, Book 55; Bukhari, Volume 8, Book 73,Narrated by 'Abdullah bin 'Amr
 Chapter 16: AN-NAHL (THE BEE), verse 72. وَاللّهُ جَعَلَ لَكُم مِّنْ أَنفُسِكُمْ أَزْوَاجًا وَجَعَلَ لَكُم مِّنْ أَزْوَاجِكُم بَنِينَ وَحَفَدَةً وَرَزَقَكُم مِّنَ الطَّيِّبَاتِ أَفَبِالْبَاطِلِ يُؤْمِنُونَ وَبِنِعْمَتِ اللّهِ هُمْ يَكْفُرُون