Codename One has an optional SaaS approach so this might (and probably will) change in the future to accommodate improved architectures. Notice that Codename One also provides an option to build offline which means corporations that have policies forbidding such cloud architectures can still use Codename One with some additional overhead/complexity. It also means you can use it for free without ever working with the build servers.
Currently on Android the standard Java code is executed as is. Java 8 syntax is translated using retrolambda for all platforms when its used. This allows it to be compatible to all Android versions as well as other ports.
On iOS Codename One built & open sourced ParparVM which is a very conservative VM. ParparVM features a concurrent (non-blocking) GC and it is written entirely in Java/C. This effectively means that an xcode project is generated and compiled on the build servers so its effectively as if you handcoded a native app and thus "future proof" for changes made by Apple. E.g. with recent 64bit and bitcode changes to iOS builds ParparVM needed no modifications to comply with those changes.
In the past Codename One used XMLVM to generate native code in a very similar way but the XMLVM solution was too generic for the needs of Codename One.
iOS builds are compiled and signed on Macs in the cloud using xcode (the official Apple build tool). This makes them compatible with current/future changes from Apple and allows developers to use Windows/Linux while targeting iOS. You can read more about the compatibility of ParparVM to iOS here.
In the past Codename One supported Windows Phone using a C# translator that relied on XMLVM but it was not an ideal approach. Notice that the XMLVM backend that translates to C# is very different from the one that was formerly used to translates to iOS. Codename One chose to discontinue that old backend as it wasn't as powerful as the new UWP backend and doesn't match Microsofts goals moving forward and focusing on UWP (Universal Windows Platform).
For Windows 10 desktop and Mobile support Codename One uses iKVM to target UWP (Universal Windows Platform) and has open sourced the changes to the original iKVM code in the Codename One github repository.
Notice that UWP builds are done on a Windows 10 machines in the cloud thus allowing developers to use Mac/Linux or older versions of Windows when building native windows apps...
JavaScript build targets which are available on the enterprise level subscribers use TeaVM to do the translation statically. TeaVM provides support for threading using JavaScript by breaking the app down in a rather elaborate way. To support the complex UI Codename One uses the HTML5 Canvas API which allows absolute flexibility for building applications.
For desktop builds Codename One uses javafxpackager, since both Macs and Windows machines are available in the cloud the platform specific nature of javafxpackager is not a problem.
What makes Codename One stand out is the approach it takes to UI where it uses a "lightweight architecture" to allow the UI to work seamlessly across all platforms and be developed almost entirely in Java. It is augmented by the ability to embed "heavyweight" widgets into place among the "lightweights". You can learn more about this in this blog post. Notice that at this time peering is undergoing some improvements and now supports more elaborate usages such as layering.
A lightweight component is written entirely in Java, this allows developers to preview the application accurately in the simulators & GUI builder.
Codename One achieves fast performance by drawing using the native gaming API's of most platforms e.g. OpenGL ES on iOS.
The core technologies behind Codename One are all open source including most of the stuff developed by Codename One itself e.g. ParparVM but also the full library, platform ports, designer tool, device skins etc. You can learn more about using the Codename One sources here.
FYI Shai Almog, the author of this answer, is the CEO of Codename One.
