There were two main ciphers here, the Enigma which is quite well known and mentioned by /u/Archek here. The Enigma machine was relatively small and portable hence its use on U-boats, etc.

The other was Tunny which was a teletype based code used more for high-volume command level messages (no pressing of buttons and noting down lights). Tuny is famous because this is where the first programmable electronic computer was built by Tommy Flowers to do the decryption.

The company that developed this was C. Lorenz AG, an electrical manufacturer based out of Berlin. Interestingly in 1930, a subsidiary of ITT, Standard Elektrik bought them out. Now I'm not quite sure what that meant in wartime but before the war they were doing fundamental work on radio such as developing the ferrite rod antenna. They also started looking at aspects of radio communication for the military and the development of the telex. Despite having visited their follow on companiy in the nineties, I do not know who actually invented the SZ40/SZ42 though so it is probable that it was a team.

The Lorenz cipher used a Vernam technique but instead of a completely random key, which is impractical, they created a very complicated pseudo-random key generated. This would be initialised with the 12 character message key and then would emit a stream of pseudo-random numbers that would be added to the message key. Nothing wrong here. Many ciphers have used a similar principle. However, if correct procedures were not followed, they could be decrypted.

Note that due to the nature of the operations and the proximity of the Soviets, post war the headquarters and the cryptography, was moved to Zuffenhausen in Stuttgart (it became SEL in the fifties, which survived for many years). Note that although the company moved to Stuttgart, enough people remained in what became the DDR that several machines appeared based on similar concepts (the same happened with Enigma machines) until the seventies. This is why the knowledge on how to break these codes became so important and secret. Note that even the West Germans would have not realised that the Lorenz codes were compromised until the seventies and they too continued to produce machines that worked similarly.

Btw, there is an excellent exhibition on this at Bletchley Park which shows the successors to the wartime technology that was broken.

Leo Marks worked on British secret codes during the war. His book Between Silk and Cyanide is an excellent semi-autobiographical account of this aspect of the war.

The British apparently made a lot of use of one-time pads, printing the pads onto silk sheets that were then sewn into the liners of agents' suits, so that they would not be apparent even under a full search and pat-down. The agents would use scissors to cut off rows from the pad and use that to encode their message, before burning it. Not only was this cryptographically secure, but it did not compromise any cypher systems or code books if the agent were captured.

However, if the agents were captured and forced to turn, or killed and then impersonated by a Nazi using their captured pad and Morse code radio set, it would be extremely difficult to tell. So they also had a lot of interesting and subtle methods for signalling and detecting which agents were compromised. This happened quite often in the continental resistance movements, so it was quite an active and fascinating part of the war.

I decided to focus specifically on the Enigma machine, since I figured that's what most people know about.

The Enigma machine was created by Arthur Scherbius. Finding information on him...was not easy.

He was, as far as I can tell, a Berlin engineer. No article I've seen has mentioned anything about him working with others. I have noticed that his machine failed to sell commercially, but was adopted by the German Navy in 1926. The German army followed 2 years later. In the 1930s, all nations began to adopt their own cipher machines as well.

An article in the Spectator, which calls itself the oldest published newspaper in English, notes how hardly anyone knows about Scherbius.

I did manage to find that he filed a patent for the machine in 1918, and intended it as a way to provide secure communications for the banking industry, according to a Daily Mail article (not the best of sources).

The Polish American Journal says that he took out the patent in 1919, and created a company in 1923, to market his invention. Only Germany was interested in it, apparently.

Honestly, the number of sources I've seen that just say that is ridiculous. I can't believe how little I can find on this Scherbius fellow, and what he wanted with the codes/how he intended to use them.

I wish I could find more, but I have no idea where to look at this point.

Sources:

The History of Cryptography: How the History of Codebreaking Can Be Used in the Mathemathics Classroom with Resources on a New CD-ROM Simon Singh Mathematics in School, Vol. 32, No. 1, History of Mathematics, Codes and Crytography (Jan., 2003), pp. 2-6

Codebreaking in World Wars I and II: The Major Successes and Failures, Their Causes and Their Effects David Kahn The Historical Journal, Vol. 23, No. 3 (Sep., 1980), pp. 617-639

Riddle of the enigma. (2005, Aug 09). Daily Mail

Clowes, F. W. (2004, 06). How the poles broke the nazi code. Polish American Journal