In computer-speak, a user interface (abbreviated UI) is the means by which a program communicates with its user. It includes graphics, by which the program displays important aspects of its internal state to a user, and also mouse and keyboard actions, by which the user commmunicates to the program.
This page describes EU2's user interface. Note that all articles at the wiki that relate to the user interface should be tagged with the user interface category, as is this article. You can select the link there to see all the pages in the category.
EU2 usually runs in Windows' full-screen mode, although it is also possible to set it up to run in a window. Its screen area is divided into four main subareas: the main map, the status bar (at the top), the controls window (on the left), and the message window (at the bottom). The controls window contains many buttons (discussed in the controls window article), as well as the information screen area, and the mini-map.
You control EU2 with both keyboard and mouse, although the interface is mostly mouse-driven. When an article talks about "clicking on" something, or "selecting" something, it means that you move the mouse over it, and left-click on it.