Okay.
First of all, you have to know what size of unit you’re taslking about. The 101st Airborne is a division, which is a very, very large unit - about 10000-20000 soldiers, depending on the country and type of force.
The U.S. Army is mainly organized by divisions. The numbers of division are sort of sequential; it’s not that there’s 101 Airborne divisions, it’s that there’s 101 divisions in total; they started numbering divisions and it was #101, and it’s also airborne, so of course it’s the 101st Airborne. Think of it as being short for “101st Division Of the Army, an Airborne Division.” There’s also a 10th Mountain Division (light infantry) but it was just the 10th division they named, not the 10th division of light infantry.
Of course, it’s not quite that simple… for instance, there are THREE 1st Divisions in the U.S. Army. During WWII they started renumbering with armored divisions, so you actually could count 1,2,3 with armored divisions. But the airborne divisions were just numbered in the line of infantry divisions. And there aren’t 101 divisions or even close to it. Some of the numbers are old and used as tradition, like the 1st Cavalry. The numbers from 26 to 50 are all reserve units, some divisions were created in WWII and don’t exist anymore… you get the idea. The U.S. Army today has “only” 10 divisions of regular troops:
1st Infantry
2nd Infantry
3rd Infantry
4th Infantry
25th Infantry
1st Cavalry
1st Armored
10th Mountain
82nd Airborne
101st Airborne
There are also quite a few reserve divisions, but they are not kept at full strength.
Above the level of division you get into Corps and Armies, huge, ponderous forces made of divisions. Patton commanded the Third Army in France, a force that at the height of its size had twenty divisions or more. But corps and armies are generally ad hoc wartime organizations, with divisions switched in and out as need be.
Below divisions, unit numbering gets complicated, and at this point in a long-standing army it can be safely assumed to mean almost nothing; the sheer number of units that have been organized, moved, switched between formations and stuck around when all the previous numbered units in the sequence were dissolved defies any sort of lucid explanation.