The Soviet Union has/had the most powerful (Hydrogen) Bomb ever, the Tsar Bomb.
Why did they name it Tsar Bomb, when they had freed themself from the Tsars before, and even executed Tsar Nicholas II?
The official designation for that particular device was the RDS-220. The nickname Tsar Bomba was an appellation applied by the West, rather than the designers of the bomb (who - according to the site linked above - apparently referred to it as Big Ivan, or simply the Big Bomb).
According to the information on nuclearweaponarchive.org,
The nickname Tsar Bomba is a reference to a famous Russian tradition for making gigantic artifacts for show. The world's largest bell (the Tsar Kolokol) and cannon (the Tsar Pushka) are on display at the Kremlin.
The only reference to the device that I could find made by a member of the design team simply refers to the RDS-220 as the "Big Bomb":
"... Krushchev was already familiar with the test program, and in particular with our program to explode a device of record-breaking power, the "Big Bomb"."
(It's worth noting that "Big Bomb" is the English translation from the original Russian. It would be interesting to know if there is an alternative translation if anyone has access to a copy in Russian).
In any event, since Sakharov headed the design team, the fact that he doesn't use the term Tsar Bomb in his memoirs would seem to support the idea that it is a western designation for the device.
In Russian Language the word "Tsar" has also another, non-literal meaning.
Examples are: "Tsar-pushka" (king of the guns), the largest (in caliber) existing gun, and "Tsar-kolokol" (king of the bells), the largest bell in the world. Both the gun and the bell are currently in the publicly accessible part of the Kremlin, (and were there during the last two centuries), they are popular tourist attractions, many Russians have seen them, and all Russians know them. The analogy is clear: Tsar bomb was the largest (most powerful) bomb ever exploded. Anyway, it seems probable that the name "Tsar bomba" is of Russian (not US) origin.
Remark. Neither the gun nor the bell were ever used (the bell cracked when they were casting it). Hopefully this will apply to the bomb as well.
1. (As already mentioned in other answers/comments) The literal meaning of Russian
Tsar Bomba is
The King of the Bombs. Just that, not
The Bomb of a/the King/Tsar or something like that (i.e. there's no specific "royalist" connotation there or any ideological connection to a monarchy as a form of government in general).
Tsar Bomba neither was the official name of the device nor it was an unofficial nickname used by its designers. This nickname came in use much later (80s?) and it's not even clear if the Soviets started to use this form first or it came from the West.
(For more details on the (nick)names of the device and origins of the "Tsar Bomb" nickname, please see other answers here).
And finally even if such name was (but yet again it wasn't) invented by the bomb designers, there would be nothing strange:
3. As already mentioned the Russian word
tsar is basically an equivalent of the English word
king having almost the same wide connotations/meanings (probably even wider). I.e. just like eating
Burger King does not make you a royalist in your place, use of
Tsar word did not make you a royalist in the USSR.
tsar and its derivatives are quite common in Russian and, unless stressed specifically (e.g. as in "Tsarist Regime" or "Tsarist Henchmen"), the word brings no ideological/political context.
In short: it is/was often used just as a synonym for (roughly)
etc. and so on. And it was nothing special about using this word(s) in Soviet times either (except maybe the very first years right after 1917).
4. In fact, even when it comes to the primary monarchy government meaning of
Tsar, there was no taboo or a sort of either.
Soviets did not deny or try to diminish the historical achievements of Russian Tsars (where it was applicable). Often it was quite in opposite actually (obviously stressing the achievements belong more to the people rather than to the ruler himself).
Tsar, also spelled Csar, or Czar, is derived from the Latin title for the Roman emperors, "Caesar". Usually considered by western Europeans to be equivalent to "king".
Tsar Bomba would simply mean "King of Bombs" or "King Bomb" "Emperor Bomb" etc.
There seem to be a lot of answers explaining "Tsar" as being equivalent to "King", which true-ish, but a bit of a simplification. It could be argued "Tsar" is closer to "Emperor".
The word is a Russification of "Ceasar", which of course was the patronym of the founder of the Roman Imperial system, Julius Caesar. The word "Kaiser" is the German equivalent.
Many of the Emperors after Julius Caesar were made heir-apparent via adoption, at which point they picked up the "Caesar" patronym. Eventually it just became a title the later Roman Emperors used. The Byzantines afterward took it up to designate the Emperor's heir-apparent.
The Russians in their early days did nearly all their trade with Constantinople, and looked to that city as their model for civilization. So when their own domains became worthy of the title of "Empire", the ruler naturally had to be a "Tsar".
English monarchs during their own imperial period didn't chose to take up that title, and were instead just styled as "Emperor" or "Empress". So that's probably the closest equivalent we have in English. However, we're more apt to describe the best of something as "King of the...". The only metaphorical "Emperor" I can think of in English is the Emperor Penguin.
So in this particular case "King of the Bombs" is indeed probably the best translation. Its just not a literal translation. But of course in this case it was English-speakers making a mock back-translation into Russian.
In perhaps his only positive contribution to the world, Saddam Hussein bestowed upon English the alternative phrase "Mother of all...". So in modern English you could instead argue that this is equivalent to "Mother of all bombs". Which of course the Americans do have one of - the B MOAB.