1. What is the Khilafah?
The Khilafah is the name given to the Islamic system of government. When we say that Muslims must establish Khilafah, we mean that Muslims must establish an Islamic state, with an Islamic system of government ruled by the Islamic Shariah.
The name Khilafah comes from the Arabic word "Khaleefah", which means the successor. The name "Khaleefah" is the title given to those Islamic leaders who came as successors to prophet Muhammad (pbuh). Another name for the Khaleefah is Ameer-ul-Mo'mineen, which means "leader of the faithful".
Muslims must perform the bay'ah on the Khaleefa, which means to select him as a leader and to trust him to rule by the word of Allah (Subhaanahu Wa Ta'Ala). This means that the Khaleefah must implement Islamic laws, the Shariah, in all the lands under the Khilafah. The Khilafah must include all nations in the world where the population is predominantly Muslim, and all Islamic Shariah laws must take effect in these lands with respect to all issues in the society.
2. What about Muslim countries that exist today?
Islam came as a uniting religion, meant to unite all of mankind under one rule, the rule of Allah (Subhaanahu Wa Ta'Ala). Allah (Subhaanahu Wa Ta'Ala) has warned us in many ayat in the Qur'an that we must not be seperated or divided into groups and we must all be one community and one Ummah. Therefore, the current state of affairs of the Muslim Ummah is not acceptable from an Islamic point of view. All Islamic nations must be united as one nation, one people, brothers and sisters in Islam, with one leader.
Prophet Muhammad (pbuh) also warned in many ahadith that we must not be split up into groups, and that all Muslims are like one inseperable body. In order to be true Muslims, we must follow the commands of Allah (Subhaanahu Wa Ta'Ala) and his messenger (pbuh), and we must unite ourselves into a single nation and a single Ummah under one leader.
Therefore, establishing a Khilafah uniting all Muslim nations is a religious duty on every Muslim, and it is just as important as establishing salat, fasting Ramadan and giving zakat. Not only is it a religious duty on every single Muslim, but it will also lead to economic prosperity, justice, peace and security for all Muslim communities living throughout the world.