Okay, brief overview. Bombers are part of the nuclear triad and part of what's called the Single Integrated Operations Plan or SIOP. Basically, bombers, missile subs and land based ICBMs are individual parts of this plan. And these plans range from everything including a single use on say, a military target in North Korea, to global thermonuclear exchanges with Russia. Three parts, but the only two parts that are on "alert" at the moment are the ICBM and sub portion. The bomber force is generated as world events and tensions dictate.
Now, at the moment, the bomber force has the weapons locked away in vaults typically at their base and are not uploaded except in emergencies or for exercises. There is no sitting and armed alert force like we had during the Cold War. However, they are still considered part of the SIOP system since they can be uploaded and put on alert status in a "minimum" amount of time.
Now, if (when) they go to alert status, that's sitting alert as the article suggests, they will be armed and uploaded with nuclear weapons (depending on the platform, could be gravity bombs or cruise missiles) and sitting on a five minute alert status. Long story short, crews have five minutes to get to the aircraft and be ready to taxi to launch. Could be four bombers, could be an entire squadron. Anyway...
Many times world events will dictate what targets they have pre-loaded into the aircraft. And furthermore, crews will not know these targets until they are in the air, receive coded communications on where to go and get the "go" codes sent by the National Command Authority (NCA with the President as lead). Furthermore, by law, there is what's called a two person concept. So, Trump cannot unilaterally order a nuclear strike without having at least one other civilian leader (approved by the Senate such as a Cabinet member) concur with the decision. Both individuals send this confirmation to the various command systems (like NORAD or the Looking Glass aircraft) and it gets disseminated from there. There are exceptions to this rule kind of, but only in cases of extreme emergency and overt attack.
Anyway, NCA sends the attack codes and authorization to the military command centers, they disseminate it to the individual aircraft, subs and silo crews to strike.
Now, in all honesty, this alert status is saber rattling with NK. Let's face it, there won't be a lot of targets to be had. I'd guess it likely won't be more than a base and not even a full squadron at that. Keeping these aircraft on alert is time and money consuming. Because it's not just the bombers, but the tankers and other assets that will need to be spun up.