what is the big back up on my mac

Probably Time Machine snapshots: About Time Machine local snapshots. Although they use space, they will be erased as the OS needs additional space. So their existence is not really a problem. If you wish them to all go away permanently, then see:

Time Machine Snapshots

Open the Terminal in the Utilities folder and enter or paste the appropriate command line. Press RETURN and enter your admin password when prompted. It will not be echoed. Press RETURN again.

To turn them ON: sudo tmutil enablelocal

To turn them OFF: sudo tmutil disablelocal

Note that turning them OFF will also delete all existing snapshots.

The 32 GBs is "Other" which can be almost anything that cannot fit in the other categories. This includes most of the OS X.

What is "Other" and What Can I Do About It?- Apple Support Communities