User Interface or UI is a vital part of games, since it is the part that connects the game to the gamer. The functional aspect of UI Design mainly takes into account the accessibility and the usability of the interface.
UI Flow is the flow or order of the different screens of the UI; from menus, to loading screens, to the actual gameplay screen.
Mūzaïc UI Flow Diagram
Graphical User Interface (GUI)
Another functional aspect of UI Design is in the actual layout of the elements in the HUD / GUI of the in-game screen, and this is normally represented in a GUI mock-up before it is passed on to the artists to beautify it.
Mock-Up GUI with Description
An often overlooked aspect of UI design, the audio user interface some times plays as important a part as the GUI when communicating information to the player, for example, in Halo's use of the beeping and refilling sounds to indicate low health and danger.
Since Mūzaïc was a game that was supposed to be made accessible to the blind, every aspect of the GUI had to have an accompanying AUI to make the game entire accessible, and it had a separate document solely for UI design (available here)