eM Client supports two specific kinds of Archiving features. Local automatic archiving inside the eM Client program and the Gmail archiving.

Automatic Archiving
eM Client archiving can be found in Menu>Tools>Settings>Mail>Automatic Archiving.

The way eM Client archiving works is that the program will create a special set of local folders called 'Archive'. Archive folders will be shown below all your other account folders and Archived messages will be only available in your device. If you will choose to archive messages from a synchronized account (set up with IMAP or EWS protocols) the messages will be removed from the server as well, so you will no longer be able to access them using webmail.

To make the automatic archiving actually work you need to enable it, set how often the Archiving should happen (Archive every X days), set how old the messages should be (Archive messages older than X days) and most importantly, select which accounts/folders you want to perform the Archiving on (Account Preferences).
Click the Archiving Scope field for each Account to select which folders should be included.

You can also perform the archiving manually by clicking the 'Perform Archiving' button. This function can be used even without enabling the Automatic archiving in the top checkmark (Enable Automatic Archiving).

Gmail Archive

We support the native Gmail Archive function. This is a specific function which removes the 'Inbox' label from a message in Inbox, leaving the message accessible in the 'All Mail' folder.

You can read about Gmail's native archiving and what it means on Google support. 

You can enable an 'Archive' button in the eM Client toolbars or you can use the default shortcut Ctrl+Shift+A.
The shortcut can be changed in Menu>Tools>Settings>General>Shortcuts.