The term imamate or imamah (Arabic: إمامة, imāmah) means "leadership" and refers to the office of an imam or a state ruled by an imam.
Allah (SWT) appointed all the prophets and all their successors. Prophet Muhammad (SAWA) informed about the names of his twelve successors (AS). Jaabir ibn Abdullah al-Ansaari, the pious companion of the Prophet Muhammad (SAWA) was among those who were been informed about the names of the twelve Imams after the Prophet (SAWA).
Infallible Imam informs some of his most trusted followers about the Imam after him, and that also depends on the situation. Appointment is from Allah (SWT) and informing is from the Imam (AS) about his successor.
I have addressed this question in a series of presentations that might be useful to you, in shaa Allah.
My understanding of this is that it is with respect to spiritual matters, particularly the spiritual position of Ahl al-Bayt (A) in that if some people who did not have an understanding of this heard some people talking about it openly, they might react negatively due to ignorance.
The word imām simply means one who leads. It comes from the root word 'amma, which means someone who other people follow.
The Arabic word amām means front or forward, and imām, denotes that as well, because the leader or one who is being followed is going to be in the front. The term ummah also comes from the same root word as well.
The opposite of the imām is the ma'mūm, which means the follower (ma'mūmīn -followers). The plural for imām is a'imma.
For example, the person leading prayer is called imām jamā'ah, or leader of congregational prayer. The follower/s are called ma'mūm/ma'mūmīn.
The Quran makes reference to this term, using it for righteous leaders, but also using it for evil leaders as well (like surah tawbah, 12).
Therefore, the term imām can be used in different ways, depending on its context.
It could be imām of congregational prayer. The term imām could be used as a title for a scholar, like the local imām who is an Islamic cleric. It could be used for someone who aspires to become a leader among the righteous, like the verse:
وَاجْعَلْنَا لِلْمُتَّقِينَ إِمَامًا
It could be used as an honorific title for someone who was a great leader, like Imām Khomeini.
It could be imām of the four Sunni Madhhabs, like Imām al-Shafi'i.
It could be used as a title for the chief leader, or a ruler, or a caliph.
In our Shi'i context, we avoid using the word caliph as a title for our leaders and the successors of the Prophet (s.a.w), and we refer to them as Imāms, like Imām Ali, etc.
And Allah knows best.
This Hadeeth has been narrated in many wordings but with the same meaning. This called Tawaatur Ma'nawi تواتر معنوي.
In Saheeh Muslim , V. 6, P. 22 Kitab al-Imaarah : Who dies without allegiance in his neck, will die a death of Jahiliyyah.
In Musnad Ahmed Ibn Hanbal , V. 4, P. 96: Who dies with out Imam, will die a death of Jahiliyyah.
In Kitab al-Sunnah by Ibn Abi 'Aassim , P. 489: Who dies with out Imam on him will die a death of Jahiliyyah.
The Hadeeth is also narrated in the most authentic Shia books of Hadeeth like al-Kaafi, V.1, P.276 and in many other books. It means that the meaning of this Hadeeth is Ijmaa' (unanimous) between all Muslims.
You could ask them to provide a coherent answer to who it does refer to, because it is quite a struggle to find a list of 12 names - other than the 12 Imams - that somehow relate to each other and aren't arbitrary that fit that hadith.
However, these discussions are usually unfruitful because usually people approach them with their own preconceived biases in mind, so you have to decide whether or not there will be a benefit to the debate. Sometimes it is good just to say your point once, and then move on and let them ponder it on their own.
There is a lot of material you could look at online including on this website for more details on that hadith and ways it has been interpreted by Sunnis and Shi'is.
Imamah is the divine leadership after the Prophet (SAWA) to lead and guide Muslims to the real teachings of Islam. It was mentioned in hundreds of times by the Prophet (SAWA) including the greatest announcement in Islam on the Day of Ghadeer when the Prophet ordered all Muslims to give allegiances to Ali as the commander of the faithful.
Muslims are required to obey the Prophet (SAWA) as Allah (SWT) says in Quran : Who obeys the Prophet has in fact obeyed Allah.(Sura Annisaa', verse 80).
There are hundreds of authentic narrations in Sunni books about this most authentic prophetic order
The Prophet (SAWA) has said: Whenever you mention my name as Prophet, mention Ali as Ameerul Mo'mineen.
The name of Ali (AS) was announced in the Adhan immediately after the declaration of Ghadeer by Bilal and Salmaan and other Sahabah in the presence of the Proohet (SAWA) and he approved it. This has been narrated even in Sunni books like (Al-Salafah Fi Amr al-Khilaafah by al-Maraaghi, page 36).
The enemies of Ahlul Bayt removed and banned the name of Ameerul Mo'mineen Ali from the Adhan.
The status and knowledge of the Prophet Muhammad ( SAWA) is the greatest above all the Prophets. His Progeny are from him as Allah says in Quran ( A Progeny (of the Prophet) from him and he is from them (3:34).
The knowledge of the Prophet Muhammad is higher than the knowledge of all previous prophets. Since his Progeny inherited and got his knowledge, their knowledge is higher than all human beings except the Prophet Muhammad (SAWA).
The five Ineffable greatest personalities (The Prophet (SAWA), Imam Ali (AS), Fatima (AS), Imam Hasan (AS) and Imam Husain (AS) have repeatedly said in many authentic Hadeeths that all the nine following Imams are Infallible. Believing in Hadeeth al-Kisa leads definitely to believing that what ever was said by the greatest Five is the absolute truth which is strong evidence that all the twelve Imams are Infallible.
Allah, The Glorious is everywhere and He sees everything every moment as we believe and read in Quran (He is with you where you may be) Sura 57, Verse 4.
'The Prophet Muhammad and his successors are granted knowledge by Allah as and when Allah wants. Their knowledge is gifted to them by Allah. Allah's knowledge is His own and never given to Him.
Infallible Imams are the most humble servants of Allah after the Prophet (SAWA).
Imamah is mentioned in many verses in Quran including this verse (2:177). Imamah ( Divine leadership of Islam and Muslims after the Prophet) is mentioned as part and parcel in the faith in The prophets. Believing in the greatest prophet Muhammad means believing in each and every order he gave, and he clearly announced in front of thousands of Muslims ( For whom who believe in me as his master, Verily Ali is his master. ( Bukhari in al-Tareekh al-Kabeer, V.1, P.375 and V.4, P. 193) and hundreds of other Sunni books.
There are thousands of authentic Hadeeths in all main books of Hadeeth ordering Muslims to follow Imam Ali being the real successor of the Prophet Muhammad (SAWA).
Those who believe in Muhammad as a Prophet from Allah, must obey him and follow his real successors about whom he clearly said: My successors will be twelve, all of them will be from Quraish. (Saheeh Bukhari;6682, Saheeh Muslim ;3393, Musnad ibn Hanbal;3593, Tirmithi; 2149, al-Mustadrak Alal- Saheehain ;6586 and many other books of Hadeeth).