Esri provides development packages for desktop clients such as ArcGIS Engine Developer Kit (for platforms .NET, Java, COM, multi-platform C ++).
ArcGIS Engine is a collection of programmable components (ArcObjects) and developer resources that allow custom applications to create new maps or extend functionality into existing applications (even ArcMap itself).
The ArcGIS Software Developer Kit for smart mobile devices (tablets and phones) allows you to create applications for displaying maps, geocoding, geoprocessing and customizing other features through ArcGIS Server services provided.
They also provide the ability to embed ArcGIS maps and execute tasks online:
Esri provides a development kit for each of the most popular mobile operating systems in the world market, such as: iOS, Android and Window Phone.
Runtime iOS SDK
The ArcGIS API for iOS allows you to create applications that leverage the powerful features of mapping, geocoding, geoprocessing, and other custom capabilities provided by ArcGIS for Server using Objective C, and deploy those applications on iPhone, iPod Touch and Apple iPad. The API includes examples, templates and libraries of Objective C that can be used in the integrated development environment (IDE) Xcode.
Android Runtime SDK
ArcGIS API for Android allows you to create applications that use the powerful features of mapping, geocoding, geoprocessing, and other custom features provided by ArcGIS for Server using Java, and deploy to Android devices.
The API includes a plug-in for the integrated development environment (IDE) that provides a comprehensive set of tools, documentation and samples to help developers create applications using the ArcGIS API for Android.
Windows Phone Runtime SDK
ArcGIS for Windows Phone extends the reach of your GIS from the office to the field. It is included an application for Windows Phone devices that can be downloaded for free from the Market Place and lets you explore the maps available at ArcGIS Online and ArcGIS for Server.
Using the ArcGIS API for Windows Phone, you can develop applications focused on Windows Phone devices that can be deployed in your organization or to the public through the Market.