I haven’t used all of them yet, but the ones I have used have been really useful.
- SSH: ConnectBot
- File Manager + More: ASTRO File Manager
- FTP / SFTP: AndFTP
- Remote Desktop/VNC: android-vnc-viewer
- Windows Remote Desktop: Remote RDP Lite
- Internet Speed Test: FCC Mobile Broadband Test
- Ping / DNS Lookup: Ping & DNS
- IP & Port Scanner: Network Discovery
Did I miss any good, free Android IT tools?