In a general sense, it is the portion of a program that interacts between a user and an application, meaning it is what you see on the computer screen. It usually refers to "user interface," which consists of the set of operating system commands, graphical display formats, and other features designed for use on a computer or a program.
A "graphical user interface" (GUI pronounced: goo-ey) provides users with a picture-oriented user-friendly way to see what is on a computer system. A "programming interface" consists of the set of statements, functions, options, and other ways of expressing program instructions and data. In this way, an interface is also thought of as a special point of entry into the software or operating system, where programmers can work on the actual code underlying everything.
Another definition of interface is the actual connection between two applications or two hardware devices, that facilitates the exchange of data. "To interface" is to make an appropriate physical connection between two pieces of hardware so that the equipment can communicate or work together effectively.
The word has permeated into mainstream culture; as a verb "to interface" means to communicate with another person or object, as in "Did you see those two interface with each other?"