In Sitecore SPEAK is an acronym for: Sitecore Process Enablement & Accelerator Kit
Super Quick Overview of SPEAK
SPEAK is an application development framework for use within Sitecore. It's intended to build small applications / tools for business users & marketers that act as a natural extension of Sitecore.
SPEAK accomplishes this by providing pre-built components meant to accelerate development and provide a consistent UX across the envorinment
As of Sitecore 7.5 SPEAK in its 4th generation as a framework. Unless customizing business logic is required, SPEAK operates as browser-based technology leveraging the latest in JavaScript. This is a significant departure from SheerUI, which is a server event driven framework used by the traditional Sitecore UI.