In order for us to understand the importance and philosophy of mourning for Imam Hussein (a.s.), it is not amiss to take notice of the following points:
1. Paying attention to lofty virtues and traits of elect and dignified men and religious leaders:
The Holy Quran (a.s.) has emphasized through many different verses on maintaining and reviving the memory of the great men of God, noble religious leaders and prominent personalities of human history and learning lessons from them.
God, the Sublime, says in Chapter Maryam:
"And mention Ibrahim in the Book; surely he was a truthful man, a prophet." 
Then it explains Ibrahim's strong action against the idols and idolaters.
The Quran says in another verse:
"Also mention in the Book the case of Idris: He was a man of truth (and sincerity), (and) a prophet; And We raised him to a lofty station." 
The Quran speaks of the prophets who were bestowed with God's bounties and blessings.
In another verse, the Quran says:
"Commemorate Our Servant Job. Behold he cried to his Lord: "The Evil One has afflicted me with distress and suffering! (The command was given:) "Strike with thy foot: here is (water) wherein to wash, cool and refreshing, and (water) to drink. And We gave him (back) his people, and doubled their number,- as a Grace from Ourselves, and a thing for commemoration, for all who have Understanding." 
It is clear that the personal and family aspects as well as the routine activities of these noble men have not been the focus of attention in these verses and the likes. In fact, it is their character, moral virtues and constructive programs which have been highlighted and which indicate that the memory of great and elect men should remain alive and their traits and characteristics should be the focus of attention. For this reason, we see that the infallible Imams, peace be upon them, adopted different ways and methods to revive the memories and names of the martyrs of Karbala, especially the Chief of the Martyrs, Imam Hussein, peace and blessings of Allah be upon him.
...it is their character, moral virtues and constructive programs which have been highlighted and which indicate that the memory of great and elect men should remain alive and their traits and characteristics should be the focus of attention. For this reason, we see that the infallible Imams, peace be upon them, adopted different ways and methods to revive the memories and names of the martyrs of Karbala, especially the Chief of the Martyrs, Imam Hussein, peace and blessings of Allah be upon him.
2. The Shia Imams (a.s.) and Different Ways of Mourning for Imam Hussein (a.s.):
A) Holding Mourning Ceremonies:
One of the methods that the Imams used to keep the memory of the rise of Ashura alive was arranging gatherings for crying and mourning and making others cry for the tragic incidents that took place in Karbala and reminding people of those incidents on appropriate occasions.
Imam Sajjad would constantly cry and mourn for what had taken place on the day of Ashura during the years he was the Imam of the Ummah. He cried so much to the extent that he was known as and called one of the "Bakka’een" (those who cried very much).
Alqame Hadrami relates that Imam Baqir (as) would gather people in his house on the day of Ashura to cry and mourn, and the Imam himself would cry for his grandfather. There was no Taqiyyah involved and he would tell those in the house to cry for Imam Hossein and to express their condolences to each other on the day of his martyrdom. 
On one occasion Imam Sadiq told Dawood Ruqi:
"I have never drunk cold water without remembering Imam Hossein." 
It is narrated in a hadith that Imam Reza said:
"When the month of Muharram would arrive no one would see my father (Imam Kazim) smile and sadness and sorrow would take over him until the day of Ashura. The tenth day of Muharram was the day of sorrow and sadness and tragedy and he would say: Today is the day in which Imam Hossein was martyred." 
Therefore, not only would the Infallible Imams attend these gatherings themselves and cry for Imam Hussein but they would always encourage people to cry and mourn for the Imam. It has been said in a hadith that:
"Whoever cries for Imam Hossein or makes someone else cry for the Imam will be rewarded paradise, and even those who show they are sad and crying [even if they aren't because of whatever reason] will also be rewarded with paradise". 
The eulogists and poets like Kumit Asadi, Da'bal Khaza'ei and Seyyed Humairi were praised by the Infallible Imams (a.s.) for their reciting elegies and composing poems on the tragedy of Karbala.
B) Visitation of Imam Hussein's Grave Encouraged:
Commemorating the nobles and religious leaders and visiting the graves of the heroes have been a good custom, one that has been practiced by various nations. Among them, the chief of the martyrs and leader of freemen, Imam Hussein (a.s) is like a glamorous sun whose name and memory should be revived and remembered befittingly. There are so many sayings from the infallible leaders about the significance of visiting the grave of Imam Hussein (a.s) in Karbala. In fact, it can be said that those noble and divine leaders took advantage of every opportunity to encourage people to visit the luminous grave of the Chief of Martyrs (a.s). It was as though the Imams were committed to persuade people to visit the tomb of Imam Hussein (a.s.) in Karbala. Not only did this create spiritual and intellectual union with the Imam but it was also a kind of campaign against the tyrant and the unjust rulers as it has always been and is the same even today. There are different narrations from the Infallibles of which we shall mention only one:
Imam Sadiq (a.s) said:
"One who visits his shrine (of Imam Hussein) with eagerness and out of affection for the Holy Prophet (s.a.w.a.), Zahra (s.a.) and Ameerul Momeneen (a.s.), will be seated by Allah on the tablecloth of Paradise. He will dine with these holy personalities (Ahle Bait (a.s.)), while the others will be absorbed in the accounting of deeds." 
3. The Attribute of Muharram: The Month of the Victory of Blood over Sword:
The month of Muharram is a book so expensive and worthwhile. Its days and nights and hours are pages with ontological and deontological lessons. Indeed, Muharram teaches man how to live and die. By saying "We shall never accept disgrace" Imam Hussein, peace be upon him, sent a message to all human generations at all ages, times and lands to seek justice and truth and to rise up against unjust rulers and tyrannical governments and not to surrender to them.
In addition, when he was alone facing thousands of blood-thirsty criminals, he called out loud, "Is there any helper to help us…". It was as though he asked all conscious men and human generation at all times to help him. He who lost all his comrades, dear and near a few minutes earlier, he who dragged Ali Akbar, his son's body to the tent a while ago, he who saw his devout brother and the standard-bearer of Karbala, Abul Fazl al-Abbas soaked in his own blood, he who had no one with him and was certain that the enemy would not spare him and knew very well that he would meet his grandfather with thirsty lips, why did he ask for help?! Surely, he did not ask others for help to save himself. In reality, he also asked future generations for help; he wants them not to forget the fading tree of monotheism and to safeguard the blood that he and his friends shed to irrigate the tree so that the Ashura movement might become perpetual and the memory of the martyrs of Karbala be kept alive permanently and Islam be insured.
When Imam Hussain (as) was alone facing thousands of blood-thirsty criminals, he called out loud, "Is there any helper to help us…". It was as though he asked all conscious men and human generation at all times to help him. He who lost all his comrades, dear and near a few minutes earlier, he who had no one with him and was certain that the enemy would not spare him and knew very well that he would meet his grandfather with thirsty lips, why did he ask for help?! Surely, he did not ask others for help to save him. In reality, he also asked future generations for help; he wants them not to forget the fading tree of monotheism and to safeguard the blood that he and his friends shed to irrigate the tree so that the Ashura movement might become perpetual and the memory of the martyrs of Karbala be kept alive permanently and Islam be insured.
4. Lamentation, the cause of unity and success:
In order for every nation to be successful, it needs something to unify them and to bring them together. Without doubt, the most important cause of unity among the followers of the Ahlul-Bayt (a.s.) which can bring millions of people together around a single pivot is mourning for Imam Hussein (a.s.) and his devout companions.
Mourning for Imam Hussein (a.s) can help deliver man from tyranny, injustice and colonial rule. Through the history, lamentations for the martyrs of Karbala have led to major and minor uprisings and revolutions such the revolution of Tawwabeen (the Penitents), the revolution of Mukhtar who killed several enemies of the Holy Prophet's family to avenge the tragedy of Karbala. Mukhtar Thaqafi inflicted a serious blow upon the strong Amawid army in the wake of the event of Ashura.
Generally speaking, according to some experts, most of the revolutions of the world have been inspired by the revolution of Imam Hussein (a.s.) and his friends in Karbala. An example of it is the liberation of the Indian Sub-continent from the English colonial rule. Mahatma Gandhi is reported to have said,
“I have learned from Hussein how to be oppressed yet victorious. I have brought nothing new to the Indian nation; only my understanding of the wisdom of the hero of Karbala. If India wants to be a successful nation, it should follow the footsteps of Hussein bin Ali (a.s)." 
A more important and concrete example is the glorious revolution of the Islamic Republic of Iran led by Imam Khomeini (r.a) who shook and stunned the world with his charismatic leadership. And yet a recent example is the resistance of Hezbollah in south Lebanon against Zionist aggression. The fully geared Zionist army which was supported by the most modern weaponries were miserably defeated by the power of the faith and devotion of Hezbollah men.
Hence, therefore, it can be said that mourning for Imam Hussein (a.s.) can earn the Muslim Ummah dignity and pride and can help maintain the Islamic and religious culture.
We hope we are enlisted among the lovers of the Chief of the Martyrs, Imam Hussein (a.s) and his true followers and are blessed with their attention.