In Attack of the Clones, Lama Su says that 200,000 clone units were ready, and a million were on the way. Now I am assuming one unit means one clone trooper. Considering that there were a million planets in the Republic, these numbers are too small to make a difference. A million troops can't even overwhelm a single Earth-sized planet. Did the Republic use other troops or droid or something?
The military term "unit" does not mean one individual. In fact, a single unit can contain several hundred soldiers. The relevant factor is the number of people in command.
If we go by these numbers, we could be talking about 40,000,000 soldiers ready and 200,000,000 on the way. However, considering that clones are quickly produced, better trained, and far more easily controlled, we can assume that clone units can potentially contain much larger numbers of soldiers.
Lama Su was likely referring to the number of units that could be deployed immediately.
Here is Lama Su's original dialogue to Obi-Wan Kenobi from Attack of the Clones.
Given that clones can be fully grown in nearly half of the time when compared to normal humans and that the order was placed nearly ten years ago, it is very likely that Obi-Wan Kenobi was looking only at the first "batch" of clones.
What we are not provided is the number of "batches" or "generations" that had been ordered, only that a million more units were well under way.
This is reinforced through Lama Su's reference to the order. She implies that it has yet to be fulfilled and that it will be in the future. Thus, both figures provided are only initial estimates.
We are also shown clone troopers in all stages of life, indicating that the clone army is still being developed and that the final numbers are likely far greater than what is provided by Lama Su.
The number of Republic and/or Separatist planets isn't what's important -- it's the number of droids faced by the clone army. If there was no Separatist Droid Army then 200,000 clones would be more than enough for the Republic to force its will upon the Separatists. Conversely, had the Senate continued to be deadlocked over the Military Creation Act and the clone army hadn't already existed, the Separatists could have have easily defeated the Republic with its droid army. Imposing its will across the entire galaxy would be difficult for either the Republic or Separatists (the Republic struggled to control the outer regions of the galaxy even in peacetime) but whichever side had an unopposed army would have been able to depose its opponent's seat of government (e.g. Coruscant).
Wookieepedia's Legends article claims that there were 1 million B1 battle droids, 100,000 B2 battle droids, and 3,000 droidekas at the first Battle of Geonosis, but these figures are not well sourced. Assuming they are true, however, that puts the clone army at about a 5-to-1 disadvantage. Nonetheless, the clones had the element of surprise and over two hundred Jedi on their side. Furthermore, the clones are touted as much better than droids:
These advantages were apparently enough to overcome the clone army's numerical disadvantage since the clone army literally routed the droid army at the first Battle of Geonosis (in that the Separatists fled in disorder).
Building droids takes less time than growing clones, so one would think that the Separatists' numerical advantage would increase over time. However, the Separatists were not prepared for full-scale production at the beginning of the Clone Wars since the Separatists didn't even know of the existence of the clone army. Furthermore, the clones are age-accelerated so that they only take half the normal time to reach adulthood. And, again, don't forget the huge advantage offered by the Jedi.
The Clones were the elite assault force of the Republic, a military arm concentrated entirely on the purpose of fighting.
Obviously each member planet had it's own military, because they're well, planets. Each one has billions of people at least and they'd need militaries to regulate their own planetary scale conflicts. Imagine a planet is a planet..like ours. Just like we have towns-cities-states-countries and accompanying levels of conflict. For Star Wars, they add on top of that "Systems" and then "Galaxy."
The GAR and the Seperatists were Galactic scale armies, a fighting force that could fight all across the galaxy. As you'd imagine, moving a whole army across the galaxy would consume far more resources than just having 10 armies sit on a planet somewhere. I mean, think about it, every country in the world has an army, only about a dozen have an honest to god space program.
These forces were capable of subjugating whole planets because of orbital support. Think about it, all they'd need to do is pinpoint weak points in the enemy defenses, target it with a highly advanced strikeforce, and then mop up the incoming reinforcements with turbolaser or even asteroid strikes. A single space crusier could destroy many armies by just throwing a giant space rock near them.
The Kaminoans wasn't selling individual soldiers, so the soldier wasn't the unit. If it was, all you would have would be clones, and likely their armor and weapons. But when Yoda deployed them on Geonosis within hours of receiving them, they already had assault vehicles which they were piloting, implying that they were trained to do so. It's also unlikely that Yoda swung by Crazy Eddie's Correlian Cruisers en route to Kamino, swiped his Galactic Express card, and bought those vehicles on the spot.
The clones were delivered with the vehicles, which implies that they came with all sorts of other equipment, maybe even the interstellar transports themselves. So a "unit" was likely similar to a military unit, consisting of a group of clones and their heavy equipment.