Which partition do I need to format or fix in order to install Sierra or lion?

I have tried in vain to get my MacBook Air back up and running. I did a system restore or reinstall a while back and something broke. I've tried command+R and a bootable USB, there doesn’t appear to be a time machine partition or option. Which partition should I format? Can I clear them all and start from scratch? If someone could show me how to get this issue resolved so I can use my laptop again that’d be great.