Sometimes referred to as a Design Sprint, a Discovery Sprint is a five days, time-boxed activity of the team focusing on product Discovery.
Contrary to Jake Knapp, we will refer to Discovery Sprints rather than Design Sprints — because the latter appears to be a bit misleading as (too) many people will misunderstand it as focusing on UI design.
The key idea behind a Discovery Sprint is to run the complete Product Discovery in just five days — so target a problem, sketch a solution, build and test a prototype in a single week with a small, fully focused team.
The following video gives a fast-paced introduction:
https://www.youtube.com/watch?v=K2vSQPh6MCE
For an overview, check out the following illustration:
Source: https://www.thesprintbook.com/the-design-sprint
On the first day, the team aligns, agrees on a long-term goal, lists the sprint questions and then maps the problem at hand. The map here is much simpler than, say, a detailed User Journey Map. It's essentially a list of actors on the left, a final goal for these actors on the right, and several steps in between connecting these. A quick introduction is given in the following video:
https://www.youtube.com/watch?v=JTqCR84fzeg
Day 2 starts with a review of other solutions, good and bad ideas that can be borrowed from elsewhere. The idea is not so much to actually copy these but instead to spark ideas and free creativity. Later in the day, individual team members will work independently on sketching their own detailed solution — which may be rough as a wireframe but should be detailed enough to understand the key concepts.
https://www.youtube.com/watch?v=_ITJ5lAXQhg
On the third day, all sketches from the team are reviewed and critiqued, the best parts are selected and combined into a holistic storyboard which resembles a step-by-step guide for the prototype.
https://www.youtube.com/watch?v=7BKBFOOKbNo
The goal of day 4 is to build the prototype based on the above storyboard. Well, actually to fake it, i.e. to create a realistic experience and facade that can be tested later on with real customers. It doesn't have to be real code, typically Sketch, Adobe, PowerPoint or Keynote can be used for digital products.
https://www.youtube.com/watch?v=IGcwFV76t7o
The final day of the Discovery Sprint is for testing the prototype with real users. Those customers will have been recruited earlier and interviews are conducted with them. The team is watching, taking notes and towards the end of the day will analyze the results and plan next steps accordingly.