Your music files are never touched, whatever you do here. Orpheic keeps its own things apart from your music: the album database it downloads (about 6.3 GB), the covers and artist photos it fetched, your library, your playlists and your settings.
Do not remove Orpheic first. Install the new version over the old one: on Windows run the new installer, on a Mac drag the new Orpheic into Applications and replace the old one. On Linux the Debian, Ubuntu and Mint package arrives with your usual updates; with the AppImage, download the new file and use it instead of the old one (if a phone is paired, first choose Quit Orpheic in Settings, so that the old one is not still running). Everything you set up stays, and the album database is not downloaded again.
Open Settings. At the bottom, choose Remove Orpheic's data… It tells you how much it will remove, lets you keep your playlists, closes Orpheic and then removes the rest. After that, remove the program itself as below.
From Orpheic 0.9.6. Your phone and TV will need pairing again if you come back.
Settings → Apps → Installed apps → Orpheic → Uninstall. From version 0.9.6 the uninstaller asks whether to remove Orpheic's data too. The box is off, so that a later install finds everything as it was. Tick it to remove Orpheic's data; untick Keep my playlists too, to remove everything.
Removing by hand: the folder %LOCALAPPDATA%\Orpheic (paste it into the address bar of a File Explorer window), and the settings, kept in the Registry under HKEY_CURRENT_USER\Software\Orpheic. The uninstaller's box does both for you.
Quit Orpheic and drag it from Applications to the Bin. Like every Mac app, it leaves its data behind: use Remove Orpheic's data… first, or remove these by hand (in Finder, Go → Go to Folder):
~/Library/Application Support/Orpheic ~/Library/Caches/Orpheic ~/Library/Preferences/com.orpheic.Orpheic.plist ~/Library/LaunchAgents/com.orpheic.Orpheic.plist (only if Orpheic started by itself)
Debian, Ubuntu, Linux Mint. sudo apt purge orpheic removes the program and Orpheic's package source (with apt remove the source stays, and your system keeps asking orpheic.com for updates). As with every program, nothing in your home folder is removed: use Remove Orpheic's data… first, or remove these by hand:
~/.config/Orpheic ~/.local/share/orpheic ~/.cache/Orpheic ~/.config/autostart/orpheic.desktop (only if Orpheic started by itself)
AppImage. Delete the file, and the folders above.
If you moved Orpheic's pictures to another folder (Settings → Where the pictures live), that folder may hold your own files too. Remove Orpheic's data… takes only the pictures Orpheic put there and leaves everything else. If you remove things by hand, look in that folder, and in the faces folder next to it, and remove only what Orpheic put there.