While not necessarily as essential to the war effort overall to the same degree as something like aluminum, heavy water was massively important during the war, at least to the Germans.

By 1939, the German High Command had authorized German physicists to begin research into nuclear reactions, both for reactors and for bombs. The initial analysis by Warner Heisenberg, submitted on December 6th, 1939, stated that fission could lead to power production with uranium, but only if they could find a suitable moderator. A moderator is a material that slows down neutrons enough to cause a controllable nuclear reaction. Basically, it changes how much energy neutrons have, allowing them to interact with more atoms, and increasing the ease of producing a sustainable chain reaction.

Without a good moderator, larger scale nuclear research would have been much more difficult, as developing a reactor is necessary for the simplest method of enriching uranium, as well as obtaining plutonium. Consequently, Heisenberg's assertion that there were two likely moderators, graphite and heavy water, heavily focused Germany's efforts on testing.

Initial tests of graphite demonstrated that it wouldn't work, due to impurities in the test samples, so Germany's only lead on an effective moderator was heavy water. At the time, heavy water was absurdly difficult to refine, and the only source in Europe was a Norwegian electrochemical plant called Norsk Hydro, which had started refining it more as a hedge against future demand than anything else. It produced about 3 gallons a month, which was more than sufficient to satisfy the infant nuclear research community.

Consequently, the German chemical conglomerate I.G. Farben sought to acquire both the stocks of heavy water, as well as increase production.
The Norwegian company refused both.

After word of the purchase offer made it to France, the French government made a counterbid. The general manager of Norsk Hydro refused payment and gave the entire stock to the French to keep it out of German hands.
Nevertheless, production continued.

During the invasion of Norway, German forces made it a point to capture Norsk Hydro and forcibly increase production, with an estimate that 1.5 tonnes of heavy water could be extracted per year.

Coupled with thousands of tons of uranium ore in Belgium and the Belgian Congo, Germany apparently had the infrastructure to begin vigorous experimentation with nuclear reactors and nuclear refinement, potentially allowing them to get a massive leap on nuclear weaponization. This was only reinforced when they took over France and captured several key nuclear research laboratories. Fortunately, the heavy water was smuggled to England, which meant that Germany was still reliant on the slow production at Norsk Hydro.

Allied intelligence launched several raids to destroy Norsk Hydro, one of which destroyed nearly 500 kg of heavy water, but production was able to continue. Within several months of the successful raid, production resumed, and German forces had improved security to the point that additional commando raids were infeasible.
Instead, the US Army Air Forces launched multiple bombing raids on the facility, managing to cause severe damage, leading the Germans to plan to move the facility to a safer location in southern Germany.

Allied intelligence also discovered this plan, and ordered a local commando to destroy the heavy water in transit.
He succeeded, destroying what little heavy water remained in Nazi Europe, outside of extremely limited quantities in laboratories.

Fortunately, due to the need to have progressively higher concentrations of heavy water to properly extract nearly pure quantities, this meant that Nazi Germany was entirely incapable of producing heavy water on its own, at least without years of steady effort that they did not have.
Without a supply of heavy water, the German nuclear program stalled, with only an intended test reactor built by the end of the war.

Source
The Making of the Atomic Bomb by Richard Rhodes

Edit: I was wrong and apologize for using a biased source.

There were many critical strategic resources during the war, and they varied depending on the country. Both the Axis and Allied powers were heavily dependent on certain raw materials from beyond their borders, which is why so much of the early phases of the war were about controlling the seas and controlling colonies. This CIA report gives some very concrete examples of this, including but not limited to: rubber (a huge issue for the US, because Japan had captured the main rubber-producing resources of the US early in the war, and this resulted in the US developing a synthetic rubber industry), aluminum (of vast importance for airplanes), copper, lead, manganese, etc.

Of additional note is a strategic ore that was kept secret: uranium. The US committed to building an atomic bomb in 1942, and also committed to making it difficult for any future nation to get an atomic bomb. Both of these goals necessitated getting a monopoly on known uranium reserves, especially those from the Belgian Congo. Over the course of the war, the US not only secured huge stockpiles (tens of thousands of tons) of known uranium (and thorium) ores, they also secured contracts with many nations that gave them and the British exclusive ownership of said ores moving forward. The Combined Development Trust, as it was known, did this without the other countries or companies knowing the true value of the ores (uranium was not considered particularly important during the war; the ores it was often co-mingled with, like vanadium, had much more direct wartime relevance). It was only after Hiroshima that the strategic importance of the ore become publicly understood. For more on this, see Jonathan E. Helmreich, Gathering rare ores: The diplomacy of uranium acquisition, 1943-1954 (Princeton, NJ: Princeton University Press, 1986).

Just a reminder to those leaving responses: any answer to this question must provide an in-depth and comprehensive look at either an individual material/resource or important strategic resources in general. Just naming a single material and giving a few sentences of overview or a link to another page is not in line with our standards.