Actually, the meta for a long time has been 40-width divisions on offense. Paradox's combat system targets enemy formations by divisions. The problem with large numbers of small divisions is that the combat system often has them target separate enemy formations, which is highly inefficient and disperses firepower unnecessarily. One 40-width division packs the same amount of combat power as four 10-width divisions, but the 40-width division all shoots at a single target. Now, this is a largely marginal improvement, and not nearly as important as, say, stacking tremendous amounts of breakthrough, or properly balancing breakthrough and organization. But it's a real improvement.
Now, so long as your units can fit evenly into an 80-width combat space - space for initial engagement before reinforcement - you'll be fine. 10-width and 20-width formations are acceptable so long as they're constructed intelligently. 20-width formations are often better on defense than 40-width ones. 40-width, however, is ever so slightly optimal on offense.
Editing unit composition requires army experience, which you gain either through killing enemy soldiers in combat (or capturing, or however you remove them from enemy units) or through training (the "training" activity for units that increases unit experience and army experience at the cost of attrition in manpower and equipment). Army experience is not always easy to come by. Another limitation is that you don't want your divisions an awkward size that doesn't evenly divide into 80. Finally, as mentioned before, you have to actually produce the equipment for those extra battalions to use. You can say that you want an AT company in each infantry division, but if you don't actually make or capture the AT guns, that company is never going to happen.The tutorial didn't mention anything about editing compositions.What's the point? What prevents you from just maxing out the battalions?
The various statistics for units are important. Organization determines how long your units can stay in the fight, and usually comes from infantry, motorized rifle, or mechanized battalions. It doesn't matter if your armored divisions have a huge amount of breakthrough if they don't have enough org to stay in the fight long enough to win it. Hard attack is your units' ability to fight armored units, and usually comes from armored, AT, or AA battalions; in armored warfare, armor determines your units' ability to resist antiarmored attacks, and piercing determines their ability to make such attacks. Piercing higher than enemy armor values is Good, and vice versa. Soft attack is your units' ability to bleed enemy manpower, and mostly comes from infantry and artillery battalions. During any battle, attacking units have breakthrough, and defending units have defense, and the higher number of those two is better.
Unit composition philosophy revolves around creating essentially two types of units: straight-leg infantry-heavy formations that can hold territory and destroy pockets, and fast-moving armored formations that can break through enemy lines and encircle your enemies. There are several ways to accomplish this. The simplest way is to make two and only two kinds of divisions. First, 10-battalion (20-width) infantry divisions for holding territory. You can make these 7-infantry/2-artillery if you like, compromising some defense and org to get more soft attack. Second, 40-width armored divisions, usually comprised of 4 or 5 motor-rifle or mechanized battalions (to give the division enough org to fight), 4 or 5 medium tank battalions (for breakthrough and hard attack), and 4 self-propelled artillery or rocket artillery battalions (for more breakthrough and soft attack). You can compromise some of the med tanks for self-propelled AA, and you can swap artillery for armor if you really need to.
With that said, there are all kinds of other formations that you may find useful. Small (10-width) light tank divisions work very nicely for actually executing an encirclement. Break through with the big 40-width units, then set your light tanks rolling, and they will outrun anything on their way to a rendezvous deep in the enemy rear. They can also do variants of the "race to Moscow" strategy, in which the Germans on the Eastern Front unleash their light tanks to try to cap up all the Soviet victory points and force the Red Army's collapse that way. Marine, mountain, and airborne divisions are all useful for their particular missions; most people make them 10- or 20-width, depending. (Note that you will need a unit of transport planes stationed at the relevant airfield in order to execute a paradrop with your airborne units.) If concerned about large numbers of enemy tanks, you may find it useful to stiffen your straight-leg infantry with AT or AA.
These are all units that work well for highly industrialized societies. For countries that have large amounts of manpower but little industry, you may need to find alternate paths to victory. The Chinese factions, for example, work well with huge quantities of 10-width straight-leg infantry.
You can also invest in support companies, which may or may not be beneficial. The engineer company is almost always a good idea, because it improves unit entrenchment (which improves defense over time) and unit speed, both of which are Good Things. The logistics company is excellent for formations moving into rough terrain and for armored formations generally. Company arty is often nice, especially for those 10-battalion straight-leg infantry divisions to boost their soft attack a bit. The important thing to recognize with the companies is that there are trade-offs with them. They sacrifice some unit org in exchange for other stats, and they require extra equipment to fill out that you may not have.
Anyway. That's unit composition.
You also may have been suffering from enemy air superiority. I don't believe it's too much to say that offensive activity is impossible under enemy air supremacy and heavy enemy air attack. Even if you gain air superiority but don't have any bombers to exploit it, the mere possession of air superiority provides a big combat bonus to any units operating underneath that umbrella. It is not that difficult to outproduce the AI in fighters (short-range fighters are usually best outside of, say, Africa or Central Asia), and you should make them your top priority. While it isn't that difficult to outproduce the AI, human players are another matter, and high-level multiplayer games are won and lost based on air superiority. Control of the oil and rubber needed to produce large quantities of combat aircraft is absolutely crucial to long-term success. (If the enemy relies on convoys for those resources, submarine commerce raiding and/or naval bomber attacks become extremely worthwhile. Conversely, say, a Japanese player relying on Southeast Asia and Indonesia for rubber and oil had better invest in a lot of ASW, lest she face the same disaster that historical Japan did.) Synthetic materials factories provide a tremendous bonus here and should not be overlooked. For a nation like Germany that cannot rely on significant oil or rubber supplies, they are absolutely essential. But even for, say, America, which is exceedingly rich in oil, synthetic factories usually have few drawbacks. Since they have no operating costs, they are essentially free resources falling out of the sky forever.
Even if you can't rely on air superiority, constructing AA can mitigate some of the worst of the problems of enemy bombing. Province-level AA reduces the effectiveness of enemy strategic bombing, and divisional AA reduces the effectiveness of CAS. You should invest in those if you believe yourself likely to fall behind in the air at the outset of the campaign.
