Here's more or less the timeline. Khrushchev announced, in early October 1961 or so, that the USSR would detonate a 100 Mt bomb by the end of the month. This gave enough time for plenty of outcry worldwide — basically nobody thought this was a good idea, even countries that were often allies of the USSR. The UN passed a resolution pleading with the USSR not to do it. At the end of the month, the USSR did it anyway, albeit reducing the yield by 50% to avoid adding too many fission products to the atmosphere.

The initial test was detected seismographically by the French who announced it had been done. The US, to my amusement, announced that the French were wrong and that it was just a huge earthquake. Then there was other verification (perhaps radiological, I don't know) that it was a big nuclear bomb and the US agreed with the French for once.

The basic response was to decry it as a "terror bomb." (They did not have a clever name for it at this point — it was usually referred to as the "100 Mt bomb" or the "terror bomb" and one publication tried to get it dubbed the "K-Bomb" for Khrushchev but that didn't catch on. I don't know when the "Tsar Bomba" terminology got popular but I suspect the 1990s or so.) It was lamented as an avoidable outgrowth of the arms race and Khrushchev's "terroristic" policies. It played some role in increasing the demand for an atmospheric test ban treaty because of the fallout issues with such a weapon, which were (over)hyped by the USA (all of the USA's announcements assumed the weapon was 50% fission which was not true; the USA actually knew that this was not true from radiological analysis. It produced far fewer fission products than the US Castle Bravo test in 1954, and they were kept at very high altitudes and latitudes, circulating in the stratosphere for years).

The experts and analysts were also quick to argue that it was not a military weapon but a propaganda weapon. It was too large and too heavy to be an effective military tool and its area of damage could be easily replicated with a handful of smaller weapons. As you can imagine some magazines and newspapers delighted in making very colorful representations of this fact. This was used as further argument against the USSR — they were argued to be just testing the bomb to score political points, not to achieve new weapons advancements.

Behind the scenes, incidentally, the US intelligence agencies were not so sure. The CIA thought for several years that the USSR might be interested in putting 50 Mt warheads onto superheavy ICBMs. Within the USSR, Sakharov proposed making a 50 Mt torpedo of sorts that could be targeted at a port city, but was apparently rebuffed by generals who thought that was going a bit too far even for them.

JFK released a press release denouncing the Soviet test and also claimed that there was no real trick in making weapons so big, and that in fact the US had pursued such a thing themselves years previously but had decided not to do it. He was referring to a push for a 60 Mt bomb that took place during the Eisenhower administration, but Eisenhower had rejected it. Secretly, behind the scenes, the Department of Defense and Atomic Energy Commission were instructed to evaluate, on paper, whether the US should develop 50-100 Mt weapons ("very high yield" weapons is the term used — you know when the government adds a "very" they mean business!). Los Alamos thought that they could, without any further testing, develop a 100 Mt weapon that was half of the weight of the Tsar Bomba. They thought that with more testing they could perhaps slim it down even further in weight (though that would not necessarily mean a decrease in overall dimensions — for technical reasons the diameter of such a weapon, even if it was lower in mass, would be very large).

All of this happened while JFK was trying to figure out whether to sign the Limited Test Ban Treaty, which would make the possibility of refining such large weapons impossible (only relatively low-yield weapons can be tested underground), and feared the public relations fallout (so to speak) that would come from a discovery that the US was working on 100 Mt bombs. He even prohibited the DOD/AEC from doing drop testing of the casings for such bombs because it would be obvious to observers that they were exceptionally large. (Note that at this point, the US was fielding 25 Mt bombs, two to a bomber, so while 50-100 Mt may seem large for one bomb, the US was already fielding some pretty huge weapons.) When the US signed the LTBT, it effectively prohibited them from pursuing such weapons.

That's probably more information than you wanted. I gave a talk on this subject last November so I'm brimming with facts. At some point in the near future I will write the whole thing up as an article.

Here's an article from The Times. I couldn't find any editorial commentary from that newspaper, but there was a tone of condemnation and some discussion of the neutron bomb.

Our Own Correspondent. "U.S. Readiness For More Atmospheric Tests." Times [London, England] 3 Nov. 1961: 12. The Times Digital Archive. Web. 27 May 2014.

Not much to add to the fantastic response given by /u/restricteddata

The only amusing thing to add is that the unofficial Soviet name was Кузькина мать (Kuz'ka's Mother). It is derived from what Khrushchev said to Nixon in 1959:

В нашем распоряжении имеются средства, которые будут иметь для вас тяжёлые последствия. Мы вам покажем кузькину мать! (We have at our disposal means that will have for you very dire consequences. We'll show you Kuz'ka's mother!)

It's a rather silly idiom and comes off as a very juvenile threat, kind of like - we'll show you who's boss!

It's quite amusing that the largest yield device ever detonated got that nickname.