As @Ali has already mentioned the holy Prophet --peace be upon him and his household-- has stated that "Hussain is from me and I'm from Hussain". But what does it mean? It is clear that Hussain peace be upon him is from Prophet as he is the Prophet's grandson. But what about the part saying "and I'm from Hussain"? The Shia scholars have understood this statement as follows:
Islam, the religion, is founded by the holy Prophet, he is truly the father of every Muslim including the holiness Ali (Ali peace be upon him is "Qaasim Al-Jannate wan-Naar" which implies the divider of hell and heaven, so the holy Prophet peace be upon him and his family is Abul-Qaasim) and Al-Hassan and -Al-Hussain, peace be upon them all. But what about after the holy Prophet, if would the religion corrupt like the previous religions have corrupted by their corrupted scholars? The holy book alone to survive is not sufficient as the book needs a describer. So that the holy Prophet --peace be upon him and his household-- has stated that I would leave two precious things for you after me not to be misled, the book of Allah and my household, they would never separate from each other until they come to me together at Kuthar. (AFAIK even most Sunni scholars have recorder this Hadeeth this way and only Abu-Hurayreh has recorded this Hadeeth the way that reads: "the book of Allah and my Sunnah", and this is obviously wrong as Sunnah itself needs a describer exactly like the book) Therefore, the household of the holy Prophet peace be upon them are the describers of the book, and they are those by whom the religion would remain healthy. Therefore, here the holy Prophet peace be upon him is stating that "I am the founder of the religion and they ("my household" to whom all the Sunni brothers also respect very much, all introduced in name by the holy Prophet although some was never born until years after the Prophet had passed away) are the reason for it to survive, without them even my role in Islam will get lost, as is lost the role for Jesus and Moses peace be upon them, inviting people to real Christianity and Judaism totally different than what is now called Christianity and Judaism. To conclude this paragraph the religion is founded by the holy Prophet and guaranteed to remain healthy by his household.
But why in the Hadeeth of the holy Prophet --peace be upon him-- the stress is over Hussain --peace be upon him? Simply because he was the only one who found the opportunity to explicitly announce that the Muslim community is going the wrong way, rising to enjoin what is right and forbid what is wrong. He gave his blood to lit up a candle that is glowing for centuries by now. Also we. Mentioning this one can now read "حُسَيْنٌ مِنِّي وَأَنَا مِنْهُ" (Hussain is from me and I'm from him) like "الإسلام محمدی الحدوث و حسینی البقاء" (the religion is founded by Muhammad and guaranteed by Hussain).
The candle Imam Hussain --peace be upon him-- has lit up is still glowing and has lighten the way for many Muslims and non-Muslims, Shia and Sunni. What Shia Muslims do in AL-Muharram and AL-Safar, including Ashura, is to not let the lighted candle goes dark. We have many times in Ahadeeth adviced not to let this light goes off by crying for the martyred Imam, dressing in black and to do all these in public, for someone to ask why are these people doing such and be answered because their leader who was among the most faithful, honest, kind, respectful, knowledgeable, pious and ... was killed with many of his lovers and followers while they were much thirsty and hungry, by those who were claiming to be the true followers of his grandfather --who was the Prophet of Islam-- but were truly far from being Muslim.
Although there are Ahadeeth stating the way Imams themselves were doing in the loss of their family members and perhaps other afflictions, but there is no deed forbidden in this regard unless it will harm oneself. Even each culture may have its own way of that. Crying, even if is accompanied by yelling, is obviously permitted; hitting chests and slapping cheeks are also way to do this and it is very common that someone who suffers a great disaster may do this as an extreme form of crying; I have experienced it by myself that I couldn't understand what are these hitting to chest or laps for, but at a time that I was suffering a great disaster I was doing so while I was unaware of what I am doing, then I found myself and mentioned that I have done this when I was deep in suffer and sorrow. Anyway, it may depends on culture, neither obligatory nor forbidden. Only about hitting the body with sword, this is something under debate even between the Shia Scholars, but AFAIK it has been announced as forbidden by most Shia scholars.
For more information you can read here and here and other pages in the site.
And last, but by no means least, it should be stated that "Everyday is Ashura and everyland is Karbala" (کل یوم عاشورا و کل أرض کربلا) which implies anywhere around the globe wherein there are two groups of people, the good and bad people, the situation there is like in Karbala and people should look what Imam Hussain did and follow him, first put themselves in the right side then to withstand and fight against the bad guys, be it Wall-Street, the Zionist regime or else. According to this teaching of Karbala Islamic revolution in Iran won against the American backed government of Shah, and according to this teaching of Ashura Iran is withstanding the West although is under sanctions and etc., and according to this teaching people in Bahrain shouts "هیهات من الذلة" and etc. So Karbala is alive, we are living those past days, and we refresh every Muharram with all those mourning ways that I have described, Karbala should never die because if it dies out Islam would die and even Islamic countries all would be American and Zionism backed like some today are. Mournings and re-stating what has happened there every year is a way to let the candle keep lighting. You can see how many Muslims our enemies kill in Iraq or other countries ever year during Muharram doing such mournings, they want the candle to goes dark and Allah wants it to glow, even if the enemies don't like it.