A good old fashioned TO and E question. About mobile formations to boot! Alright, buckle up.

Armored divisions, being more or less on the cutting edge organizationally, changed near constantly on paper throughout WWII. Early American formations were based on observations made in France 1940 on German performance, and they reflected the tank heavy structure that the Germans themselves would come to regret and change; Early American armored divisions for example had 2 or 3 full sized armored regiments (often of 2 Medium battalions, 1 light) plus headquarters and headquarters company. By contrast, they'd only have a single Artillery regiment (of either 2 or 3 battalions) and an armored infantry regiment of 3 battalions. Nowhere near enough supporting arms to keep 200+ tanks on the attack. All early or pre war armored formations suffered from wild imbalances in tank:infantry ratios; and revision came from battle experience. Some formations, like the German 'LeichtePanzer' would disappear entirely as they were just too flawed to be saved by simple reorganization.

How many tanks comprised a "squad"? Or a platoon or company if those divisions existed in tank regiments?

A tank is a squad. A single entity led by a NCO, of a crew between 3-6 men depending on tank model. 3 (Soviets and British), 4 (Germans), 5 (Americans) tanks would comprise a Platoon, or "Troop" in Commonwealth nomenclature, and 2 to 4 of these platoons would comprise a Company or "Squadron" (Note: "Troop" and "Squadron" are one up in US nomenclature, that is, Company and Battalion, respectively. This is unique to the United States). Most companies would have a maintenance section, and Battalion and regiment would mirror this with a larger organization. Theoretically, a US Armored unit would attempt to have six medium tanks spare to replenish unrecoverable losses, but that was rarely achieved.

Also, what sort of manpower could you expect to find in an armoured unit compare to an infantry unit?

It varies; some Armored divisions were in fact quite large, most SS and the Panzer Lehr divisions, for example, were in fact plus-sized to comparable German formations. The Pz. Lehr had almost 3000 men upwards on a neighboring model '44 Panzer division in paper strength. Americans also had a further distinction with their 2nd and 3rd Armored divisions retaining a 'Heavy' organization of two full regiments of tanks compared to a 'Light' which had 3 separate battalions. In tanks and Howitzers, a US Heavy division was nearing double the strength of a Light counterpart.

The United States remains my area of expertise, and I have more sources to hand, so lets compare a 1944 Infantry Division to a 1944 Armored Division, forgive me for the potato quality pictures...

A US infantry division is on paper at around 14k personnel. A mechanized reconnaissance company, and three Infantry regiments of about 3k men a piece making up the majority of the division's front line strength. I won't elaborate further unless asked as I don't want to get too far off the main scope of the question - I am open to explain to others what you are looking at in the photo however.

A US Heavy Armored Division is actually comparable in manpower - having slightly more, but still with the 14k figure. Two armored regiments of around 3k men each, an armored infantry regiment of comparable size, and a four-company Heavy Recon battalion would make up the majority of this unit's front line strength. The plus sized reconnaissance element, the establishment of 3 permanent tactical headquarters with personnel ("Combat commands") plus the plethora of maintenance troops would help explain why an armored division seems to have more men than an infantry division.

Last question, how did the organization of armoured units compare between the combatants of WWII?

They differed fantastically, year to year and even front to front. The Germans perhaps the most frequent to change their TO and Es, out of either organizational reforms, combat losses and front-demands, or experimentation.

To provide an example, Chris Elis, in his look at the formation of the 21.PanzerDivision, traces its evolution from the 5.LightDivision; and in doing so provides us with some interesting figures. On paper, when the 5th became the 21st, its tank strength more than doubled; with the addition of another regiment. Later in the war, German armored regiments would be two battalions of two or three companies; but early on they mirrored a more typical three battalion of three company set up; or two battalions of four companies. In theory, the 21st Panzer would've therefore have had between 116 to 174 (I believe in reality it was closer to the 116 figure) tanks of all types compared to a light division's 82 tanks of all types. A full Panzer division also had a far stronger infantry regiment (at this point in the war, still at three battalions in strength) as well as a Panzerjager detachment and an organic artillery detachment. So, more battle strength in tanks, more self-dependence in support. Unsurprisingly, Light Divisions didn't last long in the Wehrmacht's organization.

On the Eastern front, the differences between a German and Soviet division would be even more pronounced, as Soviet formations were always far smaller than a German one; in theory, a full strength Panzer Division would be more than a numerical match for a Soviet Armor Corps. To put it in perspective, let's look at the TO and E of the 3.SS and a rival Soviet Armor Corps around the time of the battle of Kursk.

Here is an incomplete OOB of the 3.SS. SS Divisions were typically a bit over strength compared to Heer equivalents, but organizationally are similar enough to make a good benchmark. Using the figures from Ellis, the 3.SS would've had roughly close to 180 main battle tanks, not including the (off picture) StuG battalion and dedicated Anti-Tank battalion. Compare this to the Soviet IInd Tank Corps and we can see the differences in strength almost immediately. A Soviet Tank "Corps" would have a comparable material and manpower size to a German "Division" on paper; before battle losses. Its the biggest gap between Axis and Allied organization one can find.

Sources:

"Decision in the Ukraine"; Nipe, George M.

"After D-Day"; Carafano, James Jay

"Hell on Wheels: The 2d Armored Division"; Houston, Donald E.

"21st Panzer Division" Ellis, Chris