The use of different colours to distinguish between different branches of the armed services was not unique to the German Army during World War Two.   The Russian Army also used coloured shoulder boards after 1942 to specifiy between the different arms, and the British Army utilized coloured strips of cloth on their sleeve to likewise distinguish the different branches.

The Germans were unique, however, in applying a large (and in the end, perhaps unwieldy) number of different colours, and to a great variety of uniform components, from hats to trousers.  The most common use of Waffenfarbe, of course, was on the shoulder straps of uniforms, but even such things as Regimental standards were in the arm of service colour rather than any specific unit colours (as was the case with British Army regimental standards).

EARLY HISTORY

The use of branch-specific colours did not have a long history in any of the world's armies by the time World War Two began.  During the Great War, the Imperial Germany Army used coloured piping to represent State or Corps affiliation to a greater degree than differentiating between branches, though limited usage with that intention did occur.  

When the postwar Reichsheer was formed in 1919, a simple system of colours was established, firmly intended to represent only the different branch of service.  The official term was Waffengattungsfarbe but this was considered long and cumbersome.

WEHRMACHT

As the Army began to expand after 1935, the system become more complicated, as differing shades of the same colour began to be utilized.  The creation of new types of soldiers (such as Mountain Troops, panzer troops, and Motorized Infantry, and later on reconnaissance troops, signals troops, assault gun troops) created new problems for the system of Waffenfarbe.

The tank troops, who were an evolution of motor transport troops, adopted the rose pink branch colour, while all transport units regardless of type (ie horse or motor) adopted blue.  However, the exception to this was tank units formed from cavalry units, who adopted golden yellow instead.    The infantrymen in tank formations also adopted rose pink.  They were called Schützen Regimenter, and to distinguish them from tank troops, were supposed to wear a "s" shaped cypher on the shoulder board. In late 1939, these infantrymen were supposed to adopt grass green waffenfarbe, but this did not actually take place until 1942/1943, with the renaming of Schützen Regiments as Panzergrenadier Regiments.

During the period of expansion, Army Officials adopted the darker Green previously worn by Jäger units, and both Jäger and Gebirgsjäger (Mountain Troops) adopted a lighter shade of green (hellgrün).  Jäger troops underwent many reorganizations during the war, and the term could be applied to different types of troops.  New titles such as Füsilier also were created during the war - a Füsilier could be a soldier in a specially designated Füsilier Regiment (which was simply a standard Infantry Regiment) or it could be a soldier in a special Füsilier Battalion (which was a reconaissance battalion of an infantry or Volksgrenadier Division).  In the former case, he wore white, in the latter, golden yellow.

The creation of Panzerjäger (anti-tank) units created even more confusion; initially they were supposed to wear rose pink piping (with a"p" cypher on the shoulder strap).  The blurring of roles of armoured vehicles led to a wide variety of self-propelled artillery and guns being created during the war; the crews of some of these vehicles wore red, others wore rose pink, still others wore golden yellow, while those vehicle crews serving in an infantry regiment, for example, wore the colour of their parent regiment (in this case white).

In all cases, Waffenfarbe was determined by the soldier's battalion or regiment, not his trade.  A medic serving in an infantry regiment wore white.  If, however, he served in the medical battalion of an infantry division, he wore the cornflower blue of medical troops.

On the whole, the system of Waffenfarbe used by the Wehrmacht in World War Two was not an eminent success.  Many changes were ordered without the ability to implement them due to material shortages; strong resistance to change was often encountered (Panzergrenadiers were especially fond of the rose pink piping that linked them to the panzer troops and many, it seems, were sad to see it go), the construction of uniform parts was hampered by ignorance of the correct shades, or else inability to properly dye different items with consistency from one batch to another, and a final hindrance was the permission given to soldiers to "wear out" older colours (especially soldiers predisposed by personal taste to retain the older colour as long as possible).  Also, once the war started, the use of devices on shoulder straps became a rarity; for troops wearing the same colour shoulder boards, this created additional confusion.

Shoulder strap devices could be embroidered (in the proper waffenfarbe) but also frequently took the form of metal devices in gilt or silver colour (and on officer's shoulder boards, these devices had to be metal, not embroidered).   Sometimes a slip on cloth loop with the device embroidered to it was used in lieu of embroidering directly to the shoulder strap.

Waffenfarben chart - showing colour and shoulder strap devices (Metal and Embroidered)