An intelligent user interface (Intelligent UI, IUI, or sometimes Interface Agent) is a user interface (UI) that involves some aspect of artificial intelligence (AI or computational intelligence). There are many modern examples of IUIs, the most famous (or infamous) being the Microsoft Office Assistant, whose most recognizable agentive representation was called "Clippy".
Generally, an IUI involves the computer-side having sophisticated knowledge of the domain and/or a model of the user. These allow the interface to better understand the user's needs and personalize or guide the interaction.
Probably the earliest examples of what could be considered true IUIs appeared in the Intelligent Computer Assisted Instruction (ICAI, aka. intelligent tutoring systems) community, which arose in the 1960s and 1970s and become popular (among academics) in the 1980s. Also, in the early 1980s, as expert systems took hold in the AI community, expert systems were applied to UIs (e.g., the aptly-named "WIZARD" system). More recent IUIs, such as Clippy, were more statistically-based, using machine learning methods to decide how to tune the interactive experience to the individual user.
What constitutes "intelligent" is potentially disputable, as is what counts as an "interface", but an easy way to make this distinction is by reference to papers published in conferences dedicated to the topic, such as the International Conference on Intelligent User Interfaces, or papers related to UIs published in AI conferences, such as AAAI.