Wow... this is a HUGE topic. As 45Deere9670 mentioned, the Marines used the Code Talkers to secure their tactical communications.

I've heard (from WWII vets) Brits used Scots Gaelic speakers for much the same reasons in certain situations - tactical field communications. I've never been able to read of accounts of this, tho. There are more Gaelic speakers than Navajo speakers, but native speakers not in the combat area were just about as remote for the Germans to kidnap as the Navajos were for the Japanese to kidnap.

The Brits had a rotor-wheel machine that was similar in concept to the Enigma machine, but had several additions that increased the cryptographic strength. This was called the Typex. The Typex the Brits used in WWII (there were umpteen variants of the Typex that went clear into the early 1960's) in WWII had five rotor wheels, compared to the Enigma's three. The Typex was a bit more user-friendly in that it had a build-in typewriter-printer for the plaintext or cipher text output of the machine.

For long-haul radio circuits, the US used a rotor-based encryption system called "SIGABA," implemented in ECM Mk I and Mk II. This was upgraded during the war several times, and used at least five rotors. To my knowledge, neither the Germans nor the Japanese broke the ECM rotor machine's output. Mathematically speaking, the ECM MkII was on par with the "cryptographic strength" of the DES system with 56 bits of key input.

For the transatlantic telephone circuit that FDR and Churchill used (frequently), Bell Labs came up with an absolutely ground-breaking crypto system, called "SIGSALY." It was the first digital voice encryption unit using pulse-coded modulation, which might not mean much to non-EE's/nerds. It was implemented with something like two dozen full-height 19" rack cabinets[*] of vacuum tube circuitry on both ends of the circuit. The "keys" were "one-time pads" of random sequences on phonograph records. After the record was used for a phone call, it was broken into pieces and thrown away. To my knowledge, this link was never broken, and probably not even understood by the Germans or Japanese. The theories of encryption used in SIGSALY were based on the work of Claude Shannon, the "father of information theory" in electrical engineering.  SIGSALY was probably the most advanced system used in the war, on any side, and probably the mode advanced system until the later 1950's. The Soviets were known for using one-time pads for their field agents - in very simple form, and this was very effective. One-time pads have a downside tho - once you've sent more traffic than you have key, your system starts to have serious vulnerabilities. 


There are a couple of "big picture" things to remember in crypto systems:

1. The more you use a crypto key, the more valuable it becomes to your enemy (or whoever is trying to intercept your traffic) and you. You need to change keys frequently, and, if possible, never use the key again. With small key sizes (like the Enigma), you can brute-force attack a message if you know some plaintext in the message and the position of that plaintext. Well, in the Enigma system, guess what the plaintext was? Every good Nazi ended his communications with "Heil Hitler" and... Bob's your uncle. There you go. This was the "ah-ha!" moment that Alan Turing had to crack Enigma.

2. Most crypto systems have poor key handling. One of the problems in most non-public key systems is "how do we get a new set of keys to both ends of the communication circuit securely?" Then, once the keys are distributed, how do you keep them secure? The logistics of key distribution is what make code cipher keybooks so valuable on both sides. When the Allies captured a U Boat, they didn't just get the four-rotor Enigma machines - they got the key books and sequence of key changes. For the Nazi navy, this was a huge problem - because to get the keys changed out on the other subs would require calling them all back in within a tight time span. In the time it would take the subs/ships to come back in to port to get new key books, the Allies could crack all their naval traffic with ease, because they had an Enigma machine and the key books. One of the most complicated issues in modern crypto systems is the distribution and repudiation of key sets.

 BTW, the Enigma system wasn't the only one used by the Germans. They also had two other systems: The Lorenz Schusselzusatz and the Geheimschreiber (translation: "secret writer") T52.  These were used for high-level, strategic communications by the Nazis, and these were married to their teletypwriter systems (aka like a telex). To attack the Schusselzusatz, the Brits developed the first electronic computer, the Colossus. Enigma was more of a field/tactical cipher system, and was used by commanders back at the central command to communicate with troops in the field or at sea. 

[*] "19 inch rack cabinets" are the standard electronic/telecomm gear rack. The full-height rack is a bit under 7' tall.