This question consists of two main parts which we shall deal with separately:
Before dealing with the main question, it is necessary to give a short account of the life and personality of Zaid ibn Ali and his uprising so as to know this sect better.
The Zaidiyyah or Zaidism is a sect which emerged in the eighth century out of Shi'a Islam. Zaidis are named after Zaid ibn Ali, the grandson of Husain ibn Ali.  Although Zaid did not consider himself to be the imam after his father, his uprising to bring the oppressors to justice and to take revenge on the enemies who were involved in the crime against the Ahlulbayt (AS) on the day of Ashura led many of his followers to consider him as the fifth Imam.  Zaid ibn Ali, who was a very brave and knowledgeable man, launched his uprising in the year 121 122 A.H.[1] against the Umayyads’ injustice and oppression in Kufa during Hesham ibn Abdul Malik, the corrupt Umayyad ruler. His uprising failed and his and many of his companions were martyred.
Zaid ibn Ali’s beliefs (uprising and revolting against the unjust under any circumstances) led many of his followers to think he was an imam in spite of the fact that he himself did not believe as such. After his martyrdom, they launched uprisings against the then governments in Iraq, Iran and Yemen. They sought to establish their own Zaidi government.
Among the most well-known and most important Zaidi governments in Iran coming into being after Yahya bin Abdullah went to Daylam and which was established by Abbasids in the year 250 A.H, we can mention the uprising led by Hussein bin Ali who was better known as the Great Inviter (al-Da’ei al-Kabir).
In 284 (A.H.), Yahya bin Zaid, leader of the Hadaviyyah sect, was able to establish the Zaidiyyah government in Sa’ada of Yemen by promoting and preaching the Hadaviyyah sect. The Zaidis’ rule over Yemen went through some ups and downs until the domination of the Ottomans.  However, following the collapse of the Ottomans, the Zaidis got another chance to reestablish their government in Yemen.  The government was in their hand until 1964 when Yemen was declared as an independent republic. 
Thus, it can be said that it is important to discuss Zaidiyyah history due, mainly, to the following reasons: 
1. Anti-Umayyad and Abbasid leadership: Many of the anti-Umayyad and anti-Abbasid movements have been led by Zaidis.  Zaid bin Ali’s uprising in Kufa and Yahya bin Zaid’s revolution in Khurasan are some of the examples to give in this regard.
2. Zaidi Shiite States:  Some of the well-known Zaidi states in the history of Islam are:
1. The Idrisians’ state in Maghreb (North West Africa):  Idris bin Abdullah bin Hasan bin Imam Hasan (AS) fled to North Africa following the abortive uprising by Shahid Fakh.  Supported by a number of tribes in that region, he established the Idrisian state in the year 172. This state continued until the domination of the Fatimids over Maghrib and it came under the protection of the Fatimid government in 305 A.H. Finally, it was completely obliterated in 375 of the Islamic calendar.
2-2. The Alawite state in Tabaristan:  This dynasty started with the Great Inviter (Hussein bin Ali), one of Zaid’s grandsons in 250 A.H.  Certain regions such as Amol, Sāri and Gorgan of Iran were included in the territory of the Alawite state.  After him were the Minor Inviter (d.278 A.H), Nasir Atroush (d. 304) and the last ruler in the dynasty, Hasan bin Qasim al-Maqtul in 316. Thus, the Alawite dynasty of Tabaristan lasted for more than sixty years.
3-2. The Alawite State of Yemen: This dynasty continued for about one thousand year.
3. Zaidis have a lot in common with Ithna Ashari Shiites as they also extensively commemorate the event of Ashura and the Ghadir festivity.
4. Zaidiyyah’s jurisprudential – theological views: The Zaidis are among the well-known sects in Islamic denominations. The scholarly works authored by Zaidi scholars from the second century as of today are among the important and reliable collections in the Islamic legacy.[2]
Jarudiyyah, Sulaymaniyyah and Salihiyyah are the three main sects within Zaidiyyah. These sects have, in turn, given rise to other sub-sects such as Hadaviyyah and Qasimiyyah in Yemen and Nasiriyyah in Iran.
Zaidiyyah beliefs
Jarudiyyah, Sulaymaniyyah, Salihiyah etc. have some beliefs in common. For instance, they believe in divine justice, man’s free will and volition, imamate of Imam Ali, Imam Hasan and Imam Husein, peace be upon them.  Imamate continues after Imam Hussein (AS) through his descendants or those of Imam Hasan (AS). They also emphasize the necessity of Amr bil-Ma’ruf and Nahy Anil Munkar. (enjoining the good and forbidding from bad).
Zaidiyyah believes that invitation (da’wah) and uprising are among the necessary characteristics of an “imam” (over and above to general conditions such as justice, knowledge etc.). This is a very important point characterizing Zaidiyyah’s political thinking. Zaidis believe in the reappearance of Mahdi (AS). They negate taqiyyah (hiding one’s religion or religious beliefs).
The differences among the Zaidiyyah sub-sects are considerable.
If the aforesaid beliefs are considered some of the commonalities bringing the Zaidi sub-sects together, there are also many differences among them. Some of the differences are enumerated as follows:
Was Imam Ali (AS) designated as his successor by the Prophet (S) himself? Among the Zaidis, Jarudiyyah maintains that the Prophet (S) designated Ali (AS) as his successor. Other sects including Batriyyah or Salihiyyah do not have such a belief.
Is it permissible for a less meritorious and virtuous imam to lead the Muslim nation in the presence of a more meritorious and virtuous imam? The answers given by Zaidiyyah sects are not identical.  Jarudiyyah does not accept it whereas Batriyyah (or Salihiyyah) and Sulaymaniyyah are of the view that there would be no problem in it.
Is it allowable to have many imams at the same time?  This question was raised with greater urgency and seriousness when Zaidis had established their states in different parts of the Islamic world. It had many proponents and opponents. 
Who is the Awaited Imam? At a time when Zaidi sects believed, under social – political circumstances, in the reappearance of Mahdi, some began to apply the name to Nafs Zakiyyah (Muhammad bin Abdullah bin Hasan bin Imam Hasan) and/or to Hussein bin Qasim ‘Ayani (d.404 A.H.).  A sect named after Husein bin Qasim (one of the Alawite Imams in Yemen) came into being in Yemen and continued to exist for many centuries. It believed that Hussein bin Qasim was the very Awaited Mahdi.[3]  However, a great number of Zaidis believe that Mahdi was not born and he will appear in future.
Belief in return and special virtues of Imams: While some of the Zaidi sects such as Jarrudiyyah believe in return (raj’ah) and in the Imams having knowledge of the unseen and exclusive virtues, some other Ziadi sects  do not have such a belief. 
Among the Zaidiyyah sects, Jarudiyyah is closer to the mainstream Shia sect i.e. Shia Ithna Ashari (Imamiyyah). The Zaidi Shiites of Yemen are connected predominantly to Jarudiyyah.
The Houthi Zaidis
Sayyid Hussein al-Houthi was the former leader and head of this ethnic group in Yemen. He was Sayyid Bardruddin’s son, the spiritual leader of the Zaidis of the Sa’ada province of Yemen. His thoughts are like those of Shia Ithna Ashari. Sayyid Hussein was killed and his brother Abdul Malik al-Houthi took the leadership but even now, Sayyid Hussein al-Houthi continues to inspire his followers.
One Yemeni scholar who has recently converted to Shi’ism has been interviewed and in his interview he gives a good picture of the Houthi Zaidis.  Since his description of Zaidis is comprehensive, we deemed it appropriate to quote him here.
Dr. Isamuddin al-Imad who is originally from Yemen was Wahhabi in the beginning. As he says, he converted to Shi’ism after research and study in the Shiite teachings and beliefs.  He is one of the students of the leader of the Houthi movement, Sayyid Hussein al-Houthi.  He says that he did not see his teachers but he had been inspired by him through his audio and video cassettes.
Dr. Isam al-Imad says that Sayyid Hussein al-Houthi and his brother Abdul Malik al-Houthi are trying to persuade Yemeni Zaidis towards the real Shiite religion i.e. Shia Ithna Ashari. He asserts that according to the Yemeni government, the Ja’fari Shia is spreading itself through Zaidi Shia and it is for the same reason that during Abdul Salih, the Yemeni government had a tough stance against Zaidi Shiites trying to convince them to follow Salafis whereas Sayyid Hussein al-Houthi was trying to do quite the opposite. He was making efforts to bring them closer to Ja’fari Shia. He reiterates that Ali Abdullah Salih, the former Yemeni president was helped by the Saudi government even though he himself was a Zaidi Shia. But for some reasons, he supported Wahhabis and Salafis.
Houthis considered Sayyid Hussein al-Houthi as their intellectual and ideological father. That is the Houthis’ strategic and ideological thought derives from Sayyid Hussein’s books. Even Abdul Malik al-Houthi himself says that whatever the Houthis have is from his brother. The Yemeni government launched a crackdown on the Houthis in the year 2004. They wanted to kill Hussein al-Houthi and his students.  I have myself listened to Houthis’ cassettes and read his books. He has tried to break the Shiites’ silence. They were silent for 30 or 40 years.
We can say that Houthis are Ithna Ashari Shiites. In fact, whatever book you see in Sa’ada (Houthis’ stronghold region) are those of the Ja’fariyah. There is no Zaidiyyah book there. In libraries and homes and everywhere you find Ja’fari books only.
For the time being, the Zaidis of Yemen are supporting Houthis heartily, even though they believe that Houthis have ideologically converted to Ja’fariya school or are, at least, inclined to Ja’faris.  The war is now with the Salafists (Wahabis) and Ba’athists. Even the Ismailis are supporting the Houthis.
In reality, the Houthis are the armed shoulder of the oppressed Shiites of the predominantly Shiite populated province of Sa’ada who are fighting Saudis on the one side and the Mansur Hadi’s men on the other. Houthis in general and Abdul Malik al-Houthi in particular have been emphasizing that Sa’ada is a part of Yemen and that they never want to separate from the rest of the country.
The Houthis fought the central government many a number of times due, mainly, to their differences over the US and Israel and the fact that the government did not pay attention to the poor condition of the Sa’ada province as it carried out no reconstruction or development projects there.
Abdul Malik al-Houthi, the current leader of the Houthi movement is a man of strong character. In spite of his elder brother being alive, he undertook the leadership of the movement. Ever since 2004, he has been able to lead the party successfully thanks to the efforts made by his brother, Yahya and his colleagues in Shabab al-Momeen.
He is a very competent orator. Badruddin, his father, was very popular among Houthis; he was a leader of the Jarudi sect (one of the Zaidiyyah sects). He made use of his influence and the respect people gave him to consolidate the position of Houthis in Yemen. Houthis’ thinking being close to that of the Imamiyyah Shia has caused their enemies to accuse them of being supported by Iran.
In his book titled “Zaidiyyah of Yemen”, Badruddin al-Houthi opposes some of the current beliefs among Zaidis causing many Zaidi leaders to criticize and oppose him.  His son, Hussein al-Houthi, was under the influence of his father’s thought. Thus, he stressed on the commonalities between Zaidiyyah and Shia Ithna Ashari and this led many people to sharply object to him.[4]
Further reading:
For further readings in regards to Zaidiyyah, read the following works.

A) Books and works written on religions and Islamic denominations such as Maghalat al-Islamiyin, Melal wa Nehal, Feraq al-Shi’ah, al-Maghalat wa al-Feragh, al-Fasl, etc.
B) Books and articles written about Zaidiyyah. Some of those books are the following: al-Zaidiyyah Qira’atun fil Mashru’ wa Bahthun ‘anil Mukawwinat, Abdulah bin Muhammad Hamid al-Din, Safaa, 1424; Al-Zaidiyyah fi Mawkib al-Tarikh, Ja’far Subhani, Dar al-Adhwaa.
C) Books and works authored by Zaidi scholars: In recent years, a lot of books have been written by Zaidi writers in and out of Yemen. Some of them can be seen in Turath al-Zaidiyyah (by Sayyid Ali Musavi Nejad, Religions and Denominations Center – 1384) and in Mu’allaft al-Zaidiyyah (by Sayyid Ahmad Husseini Ashkawari, Najafi Mar’ashi Library, Qom, 1413).

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