To be able to access folders other than your Inbox, IMAP or EWS connection to the server is required. POP3 does not support individual folders.
There are two ways to access folders from AquaMail: on an “as needed” basis, or by setting certain folders to sync (keeping them up to date, just like the Inbox).

“As needed”:
Start from the title bar or the side bar of any other folder’s message list and choose “Other folders…”. This item is also present when moving messages between folders, and also, on tablet sized devices, in the account list.

When accessing folders this way, recently accessed ones will be shown in the folder view (dropdown or side bar) and the account list. They are removed from the recent list automatically, over time, or manually via a long press on a “recent” folder to clear it. To change how long the app keeps recent folders and how many, please go to Settings -> Account list.

To enable or disable the “Other folders…” entry in the account list, please go to Settings > Manage accounts > Select your account > Options > Account options > Enable "Show Other folders...".

“Setting to sync”:
You can set any folder to sync (every time the app checks mail in the account) which will also make it always visible in the account list for easy access.

Go to Settings > Manage accounts > Select your account > Folders > Tap a folder, choose how it should be synced (most likely “Sync, plain folder”).

To clear a folder’s “set to sync” setting, go back to this screen and choose “Do not sync” for a folder you don’t need to be kept up to date all the time. There is a “+” icon in the top right on this screen (“Manage folders…”) to create a server folder.

If you delete any folders that are set to sync in the app (using web mail, or another mail app), please bring up Account Options / Folders in Aqua Mail to sync its folder list with the server.