Objective-C is the primary programming language you use when writing software for OS X and iOS. Swift language introduced recently, was sculpted from Objective-C and is a huge improvement. Swift is designed to interoperate with Objective-C and provides greater & more flexible UI controls for iOS developers to build amazing apps.
The NDK is a toolset that allows you to implement parts of your app using native-code languages. It helps to create high perfomance applications with C/C++ and integrate with Java, exploit advanced android features such as graphics and sound. You can potentially increase speed of the application.
Xamarin is a mobile application development tool. Xamarin apps share code across all platforms. With Xamarin, you write your apps entirely in C#, sharing the same code on iOS, Android, Windows, Mac and more. Anything you can do in Objective-C, Swift or Java, you can do in C#. Xamarin apps are built with standard, native user interface controls.
J2ObjC is a Java to iOS Objective-C translation tool and runtime which was created to help address the problems associated with independent releases, without sacrificing the requirement for world-class, platform-specific user experiences. It's goal is to provide a GWT-like translator and runtime.