Broadly speaking, conversational UI is a new form of interaction with computers that tries to mimic a "natural human conversation." To understand what this means, we can turn to the good old Oxford Dictionary and search for the definition of a conversation:

con·ver·sa·tion

/?k?nv?r?sāSH(?)n/ noun

A talk, especially an informal one, between two or more people, in which news and ideas are exchanged.

On Wikipedia (https://en.wikipedia.org/wiki/Conversation), I found some interesting additions. There, conversation is defined a little more broadly: "An interactive communication between two or more people… the development of conversational skills and etiquette is an important part of socialization."

The development of conversational skills in a new language is a frequent focus of language teaching and learning. If we sum up the two definitions, we can agree that a conversation must be:

Now if we go back to our definition of conversational UI, we can easily identify the gaps between the classic definition of a conversation and what we define today as conversational UI.

Conversational UI, as opposed to the preceding definition:

The gaps that we identified above represent the future evolution of conversational UI. While it seems like there is a long way to go for us to actually be able to truly replace human-to-human interaction, with today's and future technologies, those gaps will close sooner than we think. However, let's start by taking a look at how human-computer interaction evolved over the last 50 years, before we try to predict the future.