A Scrum Board is a tool that helps Teams make Sprint Backlog items visible. The board can take many physical and virtual forms but it performs the same function regardless of how it looks. The board is updated by the Team and shows all items that need to be completed for the current Sprint.
Scrum teams are self-managing. Who ever does the work is expected to keep their items on the Scrum Board visible and up-to-date. Team members should hold each other to account to make sure this happens.
If your Scrum Board is in Jira, it's not the Scrum Master's job to keep Jira up-to-date.
Many teams add categories to their Scrum Boards that fit their workflow. For example, if a Team wants to distinguish between testing and Done, they might add a ‘Testing’ or ‘Verify’ column between Done and Work In Progress.
Some Teams like to post a copy of their Sprint Burndown Chart on their board so the entire Team can see their progress during Sprint.
The board is traditionally divided up into three categories: To Do, Work in Progress and Done.
If the Team is located in the same space, it is best to have a physical Task Board. The board should be located in a central area where people can easily see and access it. In many organisations, the Task Board replaces the water cooler as the place where people gather and chat. This in turn may lead to a more collaborative work space.
If the Team is spread across multiple locations, there are a number of off the shelf programs available. Virtual Task Boards often have lots of value added features that allow Scrum Inc Product Owners and Scrum Masters to create an array of metrics to better help improve the Team’s process. Some Teams use both types of boards to get the respective advantages of each.
Don't mistake Kanban for visualising work using a Scrum Board or Task Board. Kanban is a tool for optimising flow and has strict rules about how to achieve that outcome, including limiting the amount of work in-progress in order to increase throughput.
Kanban systems facilitate just-in-time production and involve pulling work rather than pushing work (onto teams and individuals). This enables Product Backlog items to be worked on when teams have ready capacity, allowing for greater flexibility in planning of work. Kanban systems are designed to allow workers to more readily identify and measure wastes within processes. In turn this improves overall productivity.
The primary functions Kanban provide are to: 
Provide work order information
Kanban (cards) provide information about what needs to be produced, in what quantity, and any directions on the where and how the work is to be produced.
Eliminate overproduction (waste)
As work is pulled from a curated Product Backlog or Sprint Backlog, only work that is required at that time will be produced by the team. (In a knowledge worker environment this is less of a risk, particularly when the Product Owner is consistently updating the Product Backlog.)
A key component of Kanban is that work is pulled by the team, not pushed onto the team. Adopting a push approach will result in more work than the team has capacity and impacting overall performance.
Support visual control
Support and encourage improvements
Kanban will naturally lend itself to have less work in progress, leading to greater awareness and visibility of issues that impede the flow of work. Flow is the movement of value through the product development system. 
7. Mitchell, I. (2017) Walking Through a Definition of Done. https://www.scrum.org/resources/blog/walking-through-definition-done
8. Lean Lexicon – Cycle Time. https://www.lean.org/lexicon/cycle-time
9. Scrum Inc. The Scrum Board. https://www.scruminc.com/scrum-board/
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