"Khan" is reasonably translated as "king", with the understanding that a khan might be under other kings/rulers in the hierarchy. "Khan" doesn't mean "supreme ruler", just "ruler" or "king". "Khan" was the usual title for tribal leaders in Mongolia (e.g., the Merkits, Keraits, Naimans, Tatars, etc. were ruled by khans). Temujin's (the future Genghis Khan) ally and possibly overlord during the early part of his rise to power, Toghrul, was khan of the Keraits. He was usually called either Toghrul Khan or Wang Khan (or Ong Khan), the latter combining the Chinese and Turkic/Mongol titles for "king". Temujin's other famous early ally (and later rival) Jamukha rose to be the khan of the Jadaran Mongols, one of the tribes within the larger Khamag Mongol tribal confederation (which Temujin became khan of in 1189).

The higher title was "khagan" (depending on the language, the "g" can be silent or almost silent). "Khagan", of obscure original meaning, was an old Turkic title, and is usually translated as "great khan", "khan of khans", "king of kings", "emperor", etc. Unlike "khan", it has the meaning of supreme ruler, and was the usual title of the rulers of major tribal confederations and steppe empires. The etymology of "khan" is also obscure, and it is possibly a contraction of "khagan".

"Khagan" was not the only title used in the region for "supreme ruler". When Temujin's enemies allied against him in 1201, they elected Jamukha as their "Gur Khan", "universal ruler". After Temujin defeated this alliance and became the supreme ruler of Mongolia, he was elected khagan and given the title Genghis Khan, as it is commonly written in English, with "oceanic khan" or "universal khan" suggested as translations. However, in Mongolian Cyrillic, his title is Чингис хаан, best transliterated as Chinggis Khaan or Chinggis Khagan. Thus, it seems that "universal khagan" would be a better translation. His successors as rulers of the Mongol empire were also given the title "khagan", and were referred to as "Ikh Khagan" (sometimes written "Yekhe Khagan"), "Great Khagan", but usually given in English as "Great Khan". (Compare the Mongolian name of the Mongol empire: "Ikh Mongol Uls", or "Great Mongol State/Nation".

When Genghis's children and grandchildren became rulers of parts of the Mongol empire, their title was usually "khan" (Batu Khan and Chagatai Khan). The exception was Hulagu, who was given the title "ilkhan", meaning "khan of the state" or "khan of the people". It is sometimes said that this title was given to him to make explicit his subordinate position to the khagan Möngke (and later the khagan Kublai). However, not having the title "khagan" was sufficient to make his subordinate position clear, and it is possible that the intent of the title was to make clear his superior position to subordinate khans.

Use of titles is often not-quite-consistent, especially when translating to various languages. Other than the not-quite-consistent English translations above, we can note that Batu Khan was also called Tsar Batu, while the Tsars of Muscovy used the title "khagan" as the Turkic equivalent of "tsar".

Was the Ilkhanate a part of the greater Khanate or a totally separate empire?

In principle, the Ilkhanate, and also Batu's khanate (The Golden Horde AKA Ulug Ulus ("Great State/Nation") AKA the Kipchak Khanate) and Chagatai's khanate (the Chagatai Khanate) were parts of the Mongol empire. In practice, they had significant autonomy from the beginning, and their independence grew. For example, Batu's support for the election of Möngke as Great Khagan came with a demand for autonomy. The other subordinate khans were hardly under control, either - the Ilkhanate and the Golden Horde were at war with each other in the mid-13th century. Kublai used force against the Chagatai Khanate for most of the second half of the 13th century to keep it subordinate (his civil war against his brother Ariq Böke over succession as Great Khagan after the death of Möngke, and his following wars with his cousin Kaidu Khan). After Kublai's death, the Great Khagan (i.e., the Yuan emperor) had no effective control over the formerly-subordinate khanates; the Mongol empire had broken up.