Graphical User Interface (GUI pronounced "gooey") enables a user to interact with a software application through graphics instead of text. The runtime (client) of Ignition is sometimes referred to as the GUI.
In scripting, "The GUI Thread" is referring to the processing power being used to draw the tables, buttons, displays, and everything else on a client. By default most event scripting is run here, but for special tasks there are ways to move the heavy work to a different area.
The Ignition Gateway is a web server, and when it's running, it is accessed from a web browser. Any computer on the same network can access the Gateway, depending on the Gateway security settings, by using the IP Address or Host Name of the computer where Ignition is installed.