Hi and welcome
Sync is short for synchronisation. In computing it means making content the same between two parties rather than timing (as in the original meaning of the word) So, if you sync a folder on your PC with a folder on your phone, when done the contents will be the same. If you sync a contact list on your PC with one on your phone ( 2 way sync) the to lists will be the same - some will be duplicated from PC to phone and some from phone to PC to acheive this.
Auto sync allows this to be done with no specific input from the user. If auto sync is off, you have to tell the devices when to sync.
Hope this helps
Roger