Since Islam is a deen, it covers all aspects of life including the political aspect. Here is an overview of the Islamic political system. In brief, religion concerns only the personal part of life whereas deen covers all aspects of life, therefore it can be called a lifestyle.
The word khilafah has been taken from the Arabic root word Khalifa meaning to follow or to succeed. When the Prophet Muhammad (S) died, Hadhrat Abu Bakr became the leader of the Muslims. He was called KhalifatulRasulullah. Even though, Hadhrat Umar (RA) assumed the title of Ameer-ul-Momineen, the title of Khalifa was also retained and in time the Muslims adopted this title, Khalifa, for their leader.
The first four khalifas were called the rightly guided khalifa, or caliphs. During their reign, Islam was fully enforced on their people. However, soon after these four khalifas, the khilafat changed into monarchy but the name of the post remained unchanged.
By the first world war, the khalifa had shifted his headquarters to Turkey. In the first world war, Turkey fought against the British and was badly defeated. Even though the British had decided to abolish the seat of caliphate, they could not do this because the nationalist party under Mustafa Kemal Ataturk abolished it and took control.
The khilafat, in reality, is a system given by Allah. Since this system has been made by Allah, it cannot contain any flaw. This system was followed by the four rightly guided caliphs. According to this sytem, the sovereignty belongs to Allah and the Islamic Shariah or Islamic law is enforced. This law is the law given in the Quran, Ahadith, decided by ijma or consensus and qiyas or analogy.
The current system in the world, capitalism, has taken its foundations from the system of khilafat. However, since people tried to change it, it lost many of its advantages. Here's an example to prove this. If a person has a specific recipe to make a cake and he modifies it, the cake will either become better or worse. However, it is for time to prove what it has become. Similarly, time has proven that while the Muslims were using the system of khilafat, they remained the superpowers for centuries. However, no other country has remained a superpower for that long because no other country has used the system of khilafat.
It is the duty of every Muslim on the face of the earth to strive for the establishment of khlifat.
Many groups have been formed to start the journey towards the khilafat. However, it has to be considered that the group must follow the Sunnah way. This methodwas the one followed by the Prophet (S). This was that the Prophet (S) first preached and propagated Islam. While this was being done, none of his followers or him (S) was allowed to defend themselves. After gathering enough support, the Prophet (S) started teasing the Quraish and Makkans by following their caravans, etc. Finally, however, the Prophet (S) attacked Makkah and established the Islamic political system.
So far, I know about three groups that have been made in Pakistan for this purpose. Jamaat-e-Ulema-i-Islami (JUI), Hizb-ut-Tahrir and Tanzeem-e-Islami.
The JUI is of the opinion that they should win the support of the people by winning the elections and then establishing the Islamic political system in the country.
The method of Hizb-ut-Tahrir is as follows:
1. Secret stage: where individuals who join the call are cultured in Islam and the da'wah.
2. Interaction stage: where the group openly calls the society to implement Islam and interacts with all levels of society in showing (intellectually) the superiority of Islam and exposing the reality of non-Islamic systems/practices. The objective in this stage is to win public opinion in favour of Islam.
3. Seeking nusrah: takig the da'wah to those who have material and other power in society, in order to win them and through that power to implement Islam comprehensively (by establishing the state).
Tanzeem-e-Islami thinks that they should gather enough support to challenge the authority of the current political system. After that, non-cooperation movements should be started and finally, the people should stand up in open revolt. The people will not be allowed to defend. This group is of the opinion that the people who will try to resist their revolt will eventually stop resisiting as all of them are in fact each other's brothers.
Personally, I think that the best and real sunnah method is that of Hizb-ut-Tahrir. I think that gaining support by elections should not be done because the system is faulty so that rules out JUI. As for Tanzeem-e-Islami, the last step is by an analogy (qiyas) analogy of Dr. Israr Ahmed, the founder of the group. Hizb-ut-Tahrir has proved its method through the sunnah.
Finally, I conclude by saying that each one of us should strive for the establishment of the Khilafat and the best way to do this is to join one of the groups which is on the journey to establishment of Khilafat. However, you should join the group or follow the way you think satisfies you the most.