Will the candidate get an email when they progress to the next stage?

This depends on how the Administrator has set up the workflow.

If an automated email has been setup to send when the candidate reaches a specific stage, then the email will be sent.

When you click the Progress button to move a candidate to the next stage, an information box will tell you if an email will be sent or not.

You can also check by looking at the workflow. Click the icon next to the candidate's App ID in the listing to show the workflow (a link is also available in the progress information box).