Yay anon! Thanks for asking the question :) ! I haven’t received a question in months!!

I would like to begin by expressing my feeling on the topic of religion. We did not live at certain times when certain influential people lived — those who shaped what we hold and read today, be it the Qur’an, Bible, Gita, or otherwise — and so develops the word faith. One must keep this in mind when reading about any religious material. Have an open mind and respect others’ beliefs, please. 
___

The following abbreviations are used in the text below:
1. (saw) stands for Sall Allaahu ʿalay-hi wa-sallam which means ‘may Allah bless him and grant Him peace’.
2. (as) stands for ʿalayhi s-salaam which means ‘peace be upon him’. 
3. (swt) stands for Subhanahu wa-ta’ala which means ‘glorified and exalted be He’
___ 

So I will break down each term in ‘Shia Imami Nizari Ismaili Muslim’ starting at the last, which is foremost important. 

MUSLIM: Those who hold the belief that there is no god but Allah (swt) and Prophet Muhammad (saw) is the final messenger of Allah. They also follow the Qur’an (the holy book of Islam) and practice the Sunnah (the life and practices of Prophet Muhammad (saw)). 

SHIA: Before the death of the Prophet Muhammad (saw), it is believed that he appointed his cousin (who also happened to be his son-in-law) as an Imam (rightful guide) to help the followers of Islam at the time. This man’s name was Ali Ibn Abi Talib (as), more commonly known as Imam Ali (as). He is the first Imam of the Shia’te al-Ali (Party of Ali), more commonly known as the Shia.

IMAMI:It is believed that before Imam Ali (as) passed away, he appointed a successor who continued to serve as an Imam of the Shia Muslims. This process of succession, known as nass, is said to have continued. Those who believe in some form of Imamate (a line of rightful Imams) are referred to as Imami Muslims. 

ISMAILI: Ismailis are those who accepted Ismail ibn Jafar (as) as their Imam rather than Musa al-Kazim (as). Those who continued Imamate from Musa al-Kazim (as) are most often known as the ‘Twelvers’ because they have a total of twelve Imams. (See visual below).

NIZARI: Ismaili Imamate continued until Imam Al-Mustansir (as) where another split took place — those accepting his son Al-Musta’li (as) as the next Imam came to be known as the Mustaali Ismaili Muslims and those who followed Nizar (as) as the next Imam came to be known as Nizari Ismaili Muslims. (See visual below). The Shia Imami Nizari Ismaili Muslims believe that Karim al-Husseini is the present Imam. Karim al-Husseini is known more commonly as Aga Khan IV, born on December 13, 1936.

___

• I refused to number the Imams (i.e. the twelfth Imam was.. …) because there is difference also on who accepts certain Imams and thus the numbers would fluctuate based on belief. 
• I also placed ‘(as)’ after Imams which are not a part of the Nizari belief because I try my best to respect others’ beliefs.

The best visual I could find to match my above explanation is demonstrated below. This representation also mentions some events which require much more detail to understand. Hopefully I have related enough so that you may understand the the surface-meaning of the words I used in my ‘About’ section.

If you or any other people have any questions, feel free to ask me and I will try my best attempt at answering them. Allahu A’lam (Allah (swt) Knows Best).