How to remotely access a Mac from my iOS app? I'd like to remotely access a Mac mini from my iPad app.   The simplest method appears to be via SSH but I've read this requires jailbreaking the iPad and I don't want that.   Is there an 'Apple approved' method to so this?

I don't currently need access via the internet, just via the local network.   Note that some changes will require password authentication; passwords can be encrypted in the iPad app to ensure security isn't compromised over the network.
I'm assuming that you're an application developer since you said "my iPad app".

SSH works over standard TCP, it does not require jailbreaking, so you could use that (Here's a library to use SSH with: https://github.com/Lejdborg/NMSSH). You'll just need to enable SSH (Remote Login) in the Sharing settings of the OS X computer. I'm not aware of any other non-proprietary services you can use that is built into a Mac to execute commands on it remotely. You could alternatively run your own service running on the Mac to remotely control it.

If all you need is file access, you can use AFP (Apple File Protocol) or SMB (Server Message Block) through File Sharing in Sharing settings in OS X. Or SFTP through Remote Login. You can find example client implementations of those online.