Khan is a pre-Islamic steppe title. It was originally used by the Pagan Mongols, and then was used by many people who were muslim (Turks. Persians) and many people who weren't (Chinese). Khan initially implied someone who had a large amount of tribal authority, but as steppe people settled down, it basically came to mean Prince or Emir Kaqan means Khan of Khans, and was reserved for extremely powerful leaders.

Emir, Sultan and Caliph are all Arab titles.

Emir basically means Prince in the Machiavellian/Pre-modern sense. IE. an Emir is a leader who controls some land, and probably has some killers on the pay roll, but isn't necessarily sovereign.

Sultan derives from the Arabic word "power" and basically means sovereign leader, emperor, or a leader who isn't under anyone else's authority Naturally Sultan implies more authority than emir. Unlike Caliph there aren't really religious connotations.

Caliph is the most coveted title and basically means "regent" in Arabic. If someone is a Caliph then they are a powerful, just ruler who assumes the mantle of the Prophet and is the Sultan of all true Muslims. Since Muslims haven't been unified AT ALL since the Rashidun (rightly guided) Caliphs, this title is extremely controversial.

For example During the 1500s both the Shia/mainly Persian Safavids and the Sunni/mainly Turkish Ottomans commanded vast empires and fought wars with each other. The leaders of both Empires could uncontroversially claim the title of Sultan, because it was a political fact, they didn't owe allegiance to anyone. Since both empires had feet in the steppe and the mid-east, they would likely have both Emirs and Khans who owed them allegiance and controlled land and men for them.

However, when the Ottomans claimed the title of Caliph this was super controversial, because they were not accepted by all Muslims as the true representatives of the Prophet Mohammad. While a loyal Safavid Shia would acknowledge that the Ottoman leader was a Sultan ("Just a fucking shitty one!!!!") they would never consider them a Caliph.

I don't know as much about the contemporary uses of the titles, but it seems that Malek (king) and Sultan are basically used interchangeably. Emir has the same meaning it always has. Interestingly Emir/Emira is a common first name in the Arab world and Khan is a common name across the steppe.

"““I, sultan of sultans, king of kings, the shadow of God who bestows the crown to the monarchs on earth, the supreme ruler of the Mediterranean and Black Seas, the Balkans and Anatolia, Azerbaijan, Damascus and Halep, Egypt, Mecca and Medina, Jerusalem, and all of the Arab dominions, and Yemen, and the sultan and the supreme king of many nations,

I am the son of Sultan Selim Khan and grandson of Sultan Bayezid Khan, and you, King Francis, are the governor of the French province."