The Interim Committee of the Manhattan Project (a very "high-level" committee that included the Secretary of War and some top scientific and military advisors) had decided, in spring 1945, that the target ought to be an "urban area" (a city) that would showcase the power of the bomb, ideally a place with a war installation surrounded by worker's houses. That meant it had to be of a size that would allow you to really damage it thoroughly (not too small geographically), and it had to be relatively undamaged already (so you could contrast it), and it had to be pretty likely to be destroyed by blast effects (which can be shielded by hills and mountains), and ideally it was "important" enough to be worth bombing.

So they gave this criteria to the people who were in charge of making up targets for the strategic bombing of Japan (which was already well underway; they were systematically destroying every major Japanese city with firebombing attacks and wanted a list of possible targets.

From this list, the Target Committee (a different committee, consisting of scientists and military men) narrowed it down, over the course of two meetings, to only a few that met the criteria in a May 1945 meeting: Kyoto, Hiroshima, Yokohama, Kokura, and Niigata. You can read the report to see what they said about each. They also noted that: "The possibility of bombing the Emperor’s palace was discussed. It was agreed that we should not recommend it but that any action for this bombing should come from authorities on military policy. It was agreed that we should obtain information from which we could determine the effectiveness of our weapon against this target."

Yokohama was firebombed not long after this meeting; it was probably already on the chopping block before the meeting took place. So that left Kyoto, Hiroshima, Kokura, and Niigata. This became the first target list, in order of preference by the military.

When the Secretary of War, Henry Stimson, found out about the list (he asked for it), he demanded Kyoto be removed from it. The military balked; they liked it as a target. Fearing they would try to add it back on, Stimson went to Truman personally to get his permission to remove it from the list. This is a complicated historical issue; you can read my article about it if you are eager for details. In any event, Truman agreed with Stimson and remove Kyoto from the list, ostensibly on the basis that it was not a legitimate target because it was not connected enough with the Japanese military war effort.

That left Hiroshima, Kokura, and Niigata. Geographically, Niigata is quite far from the other two cities. The military wanted to have backup targets for their first two targets at least, and so they quickly, on July 24th, went back and asked for another city that would fit the criteria above. They ended up with Nagasaki, which was added to the list, even though it had been partially bombed several times over the course of the war. So the final list on the strike order of July 25th was Hiroshima, Kokura, Niigata, and Nagasaki.

For the first mission, Hiroshima was the primary target, Kokura the secondary, Nagasaki the tertiary. Everything went "as planned" and Hiroshima was destroyed.

On the second mission, Kokura was the primary target, Nagasaki the secondary. The mission went wrong in several ways, and resulted in the bomber not bombing Kokura, but instead going to Nagasaki. (If you want the details, here is my article on the bombing of Nagasaki. I have also written at length about what we know about what happened with Kokura.)

So, anyway, the take-away is that the choice of targets was a pretty complicated process (that did, in fact, go "all the way to the top," at least on the Kyoto issue; this is, as an aside, the ONLY major decision about the atomic bomb that Truman personally took part in). The difference in deliberation between the process that ended up with Hiroshima being the first target of the atomic bomb, and Nagasaki being the second, is rather profound, I think.