You want to start a script, and make sure it always keeps running in the background.
First write a file org.myscript.plist
containing:
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
<dict>
<key>KeepAlive</key>
<true/>
<key>Label</key>
<string>org.myscript</string>
<key>ProgramArguments</key>
<array>
<string>/path/to/my/script.sh</string>
</array>
</dict>
</plist>
Copy or move your file into ~/Library/LaunchAgents/
and then run:
$ launchctl load org.myscript
Finally use something like:
$ launchctl list | grep myscript
if you want to verify that it is indeed running.
The script will continue to run forever, restarted if necessary, even after you login/logout of your user session.