In the spring of 1945, when the atomic bomb planners were starting to get close enough to making a bomb that they started thinking very concretely about operational parameters, they had a meeting where they came up with the criteria for what a good target would be. Their goal was, they concluded, to make a profound psychological impact on the Japanese, and the best way to do that would be to destroy a city that had not yet been bombed extensively, so as to showcase the power of the bomb. They wanted to target "urban areas" that, ideally, had some military relevance so that they could claim it was a legitimate military target, and they wanted cities that, ideally, would be best destroyed by the kind of damage an atomic bomb does and not by firebombing.
They asked the Army Air Forces planners to compile a list of potential targets using this kind of criteria and then went over them. In May 1945 they had meeting, the notes of which you can read, where they discussed the benefits of several targets, ultimately concluding that Kyoto, Hiroshima, Yokohama, Kokura, and Niigata met the criteria the best (in order of priority). They noted that the USAAF's firebombing campaign was rapidly limiting the number of good targets, and decided to request that several of these targets be preserved from further attack so that they could be saved for the atomic bombs.
Of that list, Yokohama never got "reserved" and was firebombed not long after their meeting. They did put Kyoto, Hiroshima, Kokura, and Niigata on a "reserve" list. The document linked-to above gives you their justification for the targets:
(1) Kyoto - This target is an urban industrial area with a population of 1,000,000. It is the former capital of Japan and many people and industries are now being moved there as other areas are being destroyed. From the psychological point of view there is the advantage that Kyoto is an intellectual center for Japan and the people there are more apt to appreciate the significance of such a weapon as the gadget. (Classified as an AA Target)
(2) Hiroshima - This is an important army depot and port of embarkation in the middle of an urban industrial area. It is a good radar target and it is such a size that a large part of the city could be extensively damaged. There are adjacent hills which are likely to produce a focussing effect which would considerably increase the blast damage. Due to rivers it is not a good incendiary target. (Classified as an AA Target)
(3) Yokohama - This target is an important urban industrial area which has so far been untouched. Industrial activities include aircraft manufacture, machine tools, docks, electrical equipment and oil refineries. As the damage to Tokyo has increased additional industries have moved to Yokohama. It has the disadvantage of the most important target areas being separated by a large body of water and of being in the heaviest anti-aircraft concentration in Japan. For us it has the advantage as an alternate target for use in case of bad weather of being rather far removed from the other targets considered. (Classified as an A Target)
(4) Kokura Arsenal - This is one of the largest arsenals in Japan and is surrounded by urban industrial structures. The arsenal is important for light ordnance, anti-aircraft and beach head defense materials. The dimensions of the arsenal are 4100’ x 2000’. The dimensions are such that if the bomb were properly placed full advantage could be taken of the higher pressures immediately underneath the bomb for destroying the more solid structures and at the same time considerable blast damage could be done to more feeble structures further away. (Classified as an A Target)
(5) Niigata - This is a port of embarkation on the N.W. coast of Honshu. Its importance is increasing as other ports are damaged. Machine tool industries are located there and it is a potential center for industrial despersion. It has oil refineries and storage. (Classified as a B Target)
When the Secretary of War, Henry Stimson, learned of their target list, however, he objected to Kyoto being on it (he argued that it was not a military target). After some back-and-forth Stimson convinced Truman to back him on this, and Kyoto was struck from the target list. So the new list (in priority) was Hiroshima, Kokura, and Niigata. This, however, was not ideal because they wanted another target near Hiroshima and Kokura to serve a secondary or tertiary priority target (Niigata was quite far away from them both), and so they went back to the USAAF planning people and they gave them Nagasaki, a city which had been bombed several times but not extensively, and which had some factories relating to producing weapons in them. The city was added to the list of approved targets at basically the last minute (July 25, 1945), when the bombing strike order was finalized.
For the August 6th mission the target list (in priority) was Hiroshima, Kokura, and Nagasaki. The mission went as planned and Hiroshima was bombed. For the August 9th mission, the target list was Kokura and Nagasaki (in order of priority). For various reasons the Kokura mission went poorly and there was no visibility over the city, and so the bomber went to the secondary target, Nagasaki.
The big take-away here is that there were many possible targets considered, and neither Hiroshima nor Nagasaki was actually the first choice of the military. Nagasaki was bombed really as a result of bad luck in the end, as it was not a highly-desired target (and indeed, its topography was very unfavorable to the atomic bomb and it was not nearly as damaged as Hiroshima was). The whole process was a lot more chaotic than most people imagine it to be; there was nothing "fated" about either city as a target, and nothing particularly special about either of them except that they had not yet been extensively bombed and could serve as blank "canvases" to display the destructive power of the atomic bombs.
In the aftermath, the people who dropped the bomb created all sorts of justifications as to why they "had" to drop the bombs on these cities — the military base at Hiroshima, the war factories at Nagasaki — but these are after-the-fact justifications to make it seem a little less controversial. The reality is very clear from the planning meeting documents: they destroyed the cities because they thought such an action would make the most impact on the Japanese (and the rest of the world, including, importantly, the USSR), for better or worse.