What is a single word for a ruling representative of a supreme leader who carries out his orders and at the same time has almost complete rule over the portion of the earth in which the supreme leader has placed him?
The word used in colonial times was Viceroy.
Nowadays, we use the term Vice-President. But "viceroy" means "vice-king." (Roi is French for king).
There are several other possibles, but I like plenipotentiary (if you're describing) and The Great Panjandrum (if you're lampooning) best. Whats the context?
There may be a better position of authority title, but would "proxy" be appropriate?
Move a little bit east and one might encounter the grand vizier.
How about regent?
a person who governs a kingdom in the minority, absence, or disability of the sovereign; a person who rules or reigns
Viceroy would be a British example, sometimes used for Governor-General. A Governor or Commisioner might cover a smaller area.
Pasha was an Ottoman equivalent.
How about factotum?
"steward" or "major-domo" come to my mind. However the steward may be mainly meant to manage the ruler's holdings when he/she is absent. A major-domo is entrusted to act in place of his master whethe the master is absent or not -- but the terms could be almost synonymous. In a financial sense, "factor" is an archaic word that may fit your purpose (witness the movie Rob Roy -- quote Montrose: "what have you done with my factor?"
because English has a horde of words for different systems, levels, and details of government. The British National Corpus used by wordcount.org reckons "government" as more common than "day" or "man".
You've got someone invested with power by a higher authority. You could describe them as a deputy or lieutenant, but only informally since as a title those more commonly refer to middling ranks of the military or law enforcement.
You've got someone carrying out orders. Again, agent, factor, factotum, and plenipotentiary work informally but not as titles. (If you wanted to be less pleasant, there's creature, tool, lapdog, monkey's paw, &c.)
Looking at titles, it depends on how this realm is set up.
Are you talking about a head of government?
Some supreme leaders have an officer who truly rules everything in their place because they are too young, feeble-minded, &c. That's a regent.
Some have an officer who handles day-to-day administration and effectively rules everything in their place because they are too lazy, religious, &c. That's a prime minister but there are scads of other names: chief minister, first minister, great minister, premier, chancellor, vizier, wasir, peshwa, shogun, some sultans and mayors. Presidents are usually the head of a republic but could also describe someone who presides over a council by the authority of a supreme sovereign. Similarly, there are other titles you could give people based on particular roles: elder, speaker, secretary, chairman, &c.
Or are you talking about someone who heads a portion of the realm?
The most powerful are regional viceroys who supervise groups of provinces. The head of a province is usually known as a governor but, again, there are many, many, many other names. Prince, duke, earl, count, baron, marquess, marquis, lord, marcher-lord... essentially any feudal or noble title originally referenced such an area and its holder's power over the land's inhabitants. You can use the native English equivalents or give foreign versions like pasha, aga, bey, gong, hou, zi. Imperial systems all have their own versions of governors, some with more power than others: prefect, intendant, tao-tai, vali, wali, khan, voivode, exarch, strategos, nawab, &c. &c. Theocratic states would use ecclesiastical titles, which usually approximate imperial ones but differ by religion and denomination.
Then you have truly clever eminence grises who control things without any title at all. Some are churchmen like Richelieu, others wives like Theodora or mistresses like Pompadour, others courtiers like Ji Ru. In East Asia, the demands of filial piety meant that retired emperors or dowager empresses could control the state as much or little as they liked, through commands to their son the emperor.