"Run" is an instance of an automation that runs for every schedule.
If you schedule an automation to run from the 1st of every month to the 5th of every month, then each run of the automation is called an “Automation run".
When you hit the Run Details option on an active automation - It will show the history of email triggers/Alert triggers/automation triggers that were run for a particular automation for the respective frequencies setup. When you hit the Run details option, on an active automation - it Shows the automation run.
To check the success/failure of an automation, click on the hamburger sign set against each automation and click on run details.