The Dashboard is where you can get a glance at all Stories and Tasks assigned to you as well as all activity on the Stories, Projects, and Epics that you follow. (more details on the Activity Feed here)
Clicking on Story cards, titles, or numbers will open them in a dialog where you can make edits as needed, and clicking on the checkbox on Tasks* will mark them as completed in the Story.
You can also edit multiple Stories by clicking the checkboxes in the upper-right hand corner of the Story and then selecting the Edit button that appears above.
*The Tasks that appear on the Dashboard are only from Stories that have not yet been completed.