Heeresgruppe (pl: Heeresgruppen) [Army Group]:
An organizational formation made up of a number of Armeen. The largest single German organizational formation during WWII. Usually consisted of hundreds of various units and upwards of a few hundred thousand men, all of which operated in a far ranging geographic region of the front. An example would be Heeresgruppe Afrika which controlled all units fighting in North Afrika at the time of its formation.
Armeegruppe (pl: Armeegruppen) [Army Group]:
By 1943 these were usually two or three adjacent Armeen, possibly but not always one German and one Axis-allied, with one of the Armee HQs (usually the German) temporarily placed in command over the others. An Armeegruppe was always subordinate to the local Heeresgruppe. Before late 1943, the term Armeegruppe had a less defined meaning, and could mean an Armee-sized grouping (Panzergruppe 2 was reinforced in August 1941 and was called Armeegruppe Guderian) or even a Korps-sized unit (such as Armeegruppe Felber).
Armee (pl: Armeen) [Army]:
An Armee consisted of one or more Korps, plus any attached or independent units or formations, units in reserve, and its own organic units. Armee served at the strategic level, not at the tactical. In theory, an Amree would have had between 60,000 and 100,000 men within its ranks.
Korps (pl: Korps) [Corps]:
A Korps contained one or more Divisionen, plus any attached or independent units or formations, units in reserve, and its own organic units. Korps served at the strategic/operational level, directing the actions of mainly Divisional sized units, as well as those independent Abteilungen and Bataillone that were attached temporarily to the Korps or that were organic to it. In theory, a Korps would have had between 40,000 and 60,000 men within its ranks.
Division (pl: Divisionen) [Division]:
Divisionen varied depending on the type of Division, with most containing between 1 and 4 Regimenter, plus any attached or independent units or formations, and its own organic units. Divisionen served at the operational level, both in combat and as the operational HQ for the Regimenter and attached units and formations of the Division. In theory, a Division would have had between 10,000 and 20,000 men within its ranks.
Read more about WWII German Divisional staff compositions
Brigade (pl: Brigaden) [Brigade]:
Brigaden sized units served either as independent units, as an organic part of a specific Division, or sometimes as an organic part of a Korps in place of a Division. Early in the war, many Divisionen consisted of one or two Brigaden, each consisting of a number of Regimenter along with the usual attached and organic units. Brigaden served mostly at the operational/tactical level. In theory, a Brigade would have had between 5000 and 7000 men within its ranks.
Regiment (pl: Regimenter) [Regiment]:
Regimenter consisted of a number of Abteilungen, along with any attached units or formations, and its own organic units. Regimenter served mainly at the tactical level. In theory, a Regiment would have had between 2000 and 6000 men within its ranks.
Abteilung/Bataillon (pl: Abteilungen/Bataillone) [Battalion]:
Abteilungen and Bataillone consisted of a number of Kompanien, sometimes also with other attached units or formations. These units served at the tactical level directly engaging in combat. The Abteilung was the smallest self-contained and self-sufficent combat formation. It was armed and manned to be fully able to maintain itself in combat without the support of other units. Any unit below the level of Abteilung usually did not have enough offensive or defensive firepower and support elements (Pionier, Panzerjäger, Artillery, MGs, etc), to maintain itself in a combat situation. In theory, an Abteilung organization would have between 500 and 1000 men within its ranks.
Kompanie (pl: Kompanien) [Company]:
A Kompanien consisted of a number of Züge. Kompanien served at the tactical level. In theory, a Kompanie would have had between 100 and 200 men within its ranks.
Zug (pl: Züge) [Platoon]:
A Zug consisted of a number of Gruppen. The Zug served at the tactical level. In theory, a Zug would have had between 30 and 40 men within its ranks.
Gruppe (pl: Gruppen) [Group]:
The smallest sub-unit in the German military, usually a component of a Zug.
Halb-Zug (pl: Halb-Züge) [Half-platoon]:
The result when a normal sized Zug was split into two seperate parts.
Trupp (pl: Truppen) [Troop]:
A small unit, smaller than the zug, usually of 10-20 men in size.
Kampfgruppe (pl: Kampfgruppen) [Battlegroup]:
The Kampfgruppe was an often used German combat formation that doesn't really have an equal in Allied organization, being closest in concept to that of an American task force. A Kampfgruppe could range in size from a Korps to an Abteilung or Kompanie in size. Most Kampfgruppen were usually around an Abteilung in size. The Kampfgruppe was essentially an ad-hoc organization of different arms (Some tanks, a few artillery pieces, infanry, some assault boats, a few anti-tank guns, etc), more-or-less organized temporarily for a specific operational task. These tasks could be short term or long term in nature, and they were often organized quickly in accordance with tactical and strategic situation at hand. Kampfgruppen were usually named after the person choosen to command the formation. Kampfgruppe Pfieffer is an example of a German Kampfgruppe. It was organized very quickly from elements of the 21.Panzer-Division while it was stationed in North Africa. It was named after the commander choosed to lead the unit, and it was composed of detached Abteilungen of infantry, armour, and artillery. It was used in operations over the span of a few days. The concept of the Kampfgruppe was a key and central element in WWII German tactical doctrine. Many hundreds of Kampfgruppen are known to have existed during WWII, while many thousands more were used but will probably never be known because of the nature in which they were often formed and saw combat.
Staffel (pl: Staffeln) [??]:
An elastic designation for several components under a headquarters section, these components being from section to platoon size. Often this was merely an administrative grouping, and the components were distributed to other sub-units in combat. It could either have its own small headquarters section, or one of the components' leaders could carry out a dual function.
Kolonne (pl: Kolonnen) [Column]:
An independent transportation unit, varying from company to platoon size, transporting equipment or supplies such as a bridge column (which in fact did not actually built the bridge it was transporting), or even as an light 'infantry' column (which consisted of a set number of horse-drawn vehicles capable of transporting a fixed tonnage).