The video does not include any significant audio track, perhaps understandably for a "leak" video, but it's worth emphasising that the audio experience, both in terms of input and output, is a key part of the Cortana experience.
The video shows Cortana being accessed from a Live Tile on the Windows Phone Start screen, but it will also be possible to access the feature from the standard search navigation button, with queries (search phrases) entered using the on-screen keyboard or via voice recognition.
The first half of the video also gives a good overview of the Cortana set up process, showing how the feature requires you to log-in to your Microsoft account, tell it your name, and answer a few general questions to help the service understand your character / requirements.
The video shows only one example of Cortana answering a question (weather), but it is understood that Cortana can parse natural language queries and will, where possible, provide a direct answer (i.e. smart results). Cortana can also be used to carry out certain actions on the phone, such as setting an alarm or entering a calendar appointment. It is not yet clear how extensive this integration is and whether it can be expanded by third party app and service providers.
The second half of the video shows the settings page for the new "quiet hours" functionality that allows users to specify a time period when they are not notified about incoming notifications and messages (e.g. at night, when you're sleeping). The settings make it clear that quiet hours can be automatically turned on and off (by Cortana) according to information in your calendar and that incoming calls and messages from an "inner circle" of contacts can be set to override quiet hours restrictions (i.e. calls from family members or other priority contacts).