X Toolkit Intrinsics (also known as Xt, for X toolkit) is a library used in the X Window System. More precisely, it is a library that uses the low-level Xlib library and provides a friendly (object-oriented-looking) API to develop X11 software with graphical widgets. It can be used in the C or C++ languages.
The low level Xlib library provides functions for interacting with an X11 server, but does not provide any function for implementing the graphical objects used in GUIs, such as buttons, menus, etc. Such objects are called widgets. The Xt library provides support for creating and using widget types, but does not provide any specific widget. Specific widgets are implemented by other libraries using Xt, such as Xaw and Motif.
A programmer can for example use the Xt library to create and use a new widget type such as a two-side button. Since applications typically require a number of standard widgets such as buttons, menus, etc., they usually rely on a library such as Xaw and Motif that provides such widgets, rather than implementing them directly using Xt.
Most modern toolkits such as FLTK, GTK, and Qt do not use the Xt library, preferring to use Xlib directly.