A conversational interface is any UI that imitates chatting with a real human. Right now, there are two basic types of conversational interfaces. There are voice assistants, which you talk to, and there are chatbots, which you type to.
Almost every major tech company has its own voice assistant. Apple has Siri, Google has OK Google, Amazon has Echo, Microsoft has Cortana, and so on. All of these voice assistants allow you to do things like play music, do a search, call someone, set an alarm, and more—just by speaking. Facebook has M, a human-assisted chatbot who lives within Messenger and can do anything for you from book a dinner reservation to buy you a car.
Conversational interfaces work well everywhere, including smartphones, desktops, smartwatches, and even devices without screens at all. They can integrate with services like Twitter, Facebook, or Snapchat, or run just in a text message window. Conversational interfaces also mean that every single function in an app or service no longer needs to be buried in a menu, or represented by an icon.
In the past, we would point at a symbol representing what we want a computer to do, and then it does it. For example, clicking an icon to open an app. With conversational interfaces, computers and humans can finally speak the same language.