Let's start from the cleanest view possible:

The vertical columns represent your Sprints, and horizontal rows (swim lanes) are your teams. It is possible to display between 1 to 7 columns at a time.  You can drag and drop tasks around to change both their Sprint, and assigned team. To read more about the latter, check out this page: Global (whole plugin)#Boardconfiguration

Inside each team's section you will find avatars of the members, a short summary of Remaining Estimate on all their tasks, icons representing how many tasks in each status they have in the current Sprints and a shortcut to their Agile Board (if configured in Teams). Every column can aggregate time values, but also Story Points and Epics - this in turn can be changed in Program-specific#specific-Board.

Backlogs

There are 2 backlogs total, each serves a different purpose.

Program backlog is the standard definition of a backlog. It contains all tasks that are not yet done and were not scheduled for any Sprint yet. You can drag and drop tasks to and from this backlog, and catching its border will allow you to resize it however you please. A search bar is available for your convenience - clicking the ABC icon will switch it to JQL.

Cadence backlog is separate for every column and contains tasks that were scheduled for this particular Sprint, but were not assigned to any resource nor team just yet. Tasks inside it are still a part of the mapped Sprint.

Timeline

One of the major features of this module is the visual timeline at the top. You can hide and show it on demand using the big timeline button in the navigation panel:

This view shows all of your Gantt Markers, Cadences and Program Increments. Drag and drop to position it, zoom in and out using the mouse scroll wheel. A Program Increment is basically a level in hierarchy above Cadences, a visual way to group them. It does not affect anything in JIRA. You can add them under the same menu where Cadences are added:

Double-clicking a Program Increment or Cadence on the timeline allows you to edit its details.

A single click will scroll the Board and center the view on the clicked element.

Dependencies

Another major point of the Board is having the ability to see, create and edit dependencies between tasks. To do so, click the Links button at the top and select All:

Looking familiar? Here's a standard template that many companies actually make on their walls using sticky notes and strings. Time to go digital!

You might have noticed that some links are displayed in red, and the Links menu allows you to show or hide selected types of links. Red are invalid links, ie ones that are not logically possible with the current plan. For instance, if a task linked with and end-start dependency starts AFTER the task it supposed to start before, the link will be in red unless these tasks are swapped in order.

To edit or create links, simply use drag and drop starting from the corner of a task:

Live sync

The 3rd crucial part of the Board is the live sync mode. Unfortunately, it's impossible to show it on a static image in this documentation. It's this small, unassuming icon in the menu bar:

When active (animated in blue), changes will be synced instantly. No waiting, no refreshing, no sync going on - a bit similar to how Google Docs works. Great for planning sessions and remote work!

Smart features

You will notice that some tasks above have a red calendar icon on them. This means that the task is scheduled on our Gantt in such a way, that it exceeds its Sprint. Clicking this icon will fix this (align the task to Sprint):

Likewise, if a problem with the assignee of a task is detected, the avatar will have a red exclamation mark and be clickable. There are 4 types of conflicts:

1) The assignee of this task belongs to multiple Teams different from this one. Please move this task to a different Team, or change the assignee.
2) The current assignee of this task is not in any Team. You might want to re-assign this task or add the assignee to this Team.
3) This team has no members, but the assignee of this task belongs to other Teams. You may want to move this task to this assignee's Team.
4) This team has no members, and the task has no assignee. The plugin cannot deal with this without you adding members to the Team or setting the assignee.

Once the avatar is clicked, a small fix dialog will pop up: