First, create a folder in which you’d like to move
your important emails to. We recommend being as descriptive as necessary, as well as creating a logical folder structure, in order to make locating your stored email as easy as possible in the future.

To do this, select Mailbox > New Mailbox. In the New Mailbox sheet, change the Location to On My Mac (or, choose a subfolder you’d like this new mailbox to appear within). 

Give the folder/mailbox a logical name, and click OK.

Next, select the messages you’d like to move to your new folder, and right-click on any selected message. From the resultant menu, choose Move To, then your desired destination. 



Auto-archiving is not available in Apple Mail. However, if you'd like to automate part of the process, you can set up a rule in Mail/Preferences/Rules to move messages of a certain criteria to a selected folder, and you can manually move messages from that folder to your On My Mac folder periodically (as opposed to searching through your mailbox for messages to archive).