The sync function on your Android device simply syncs things such as your contacts, documents, and contacts to certain services such as Google, Facebook, and the likes. The moment the device syncs, it simply means that it is connecting data from your Android device to the server.

Should Auto Sync be on or off?

Turning off auto syncing for Google’s services will save some battery life. In the background, Google’s services talk and sync up to the cloud.

Should I keep sync on?

I would leave it on, otherwise it could affect things like notifications and data backups. If you’re on a recent-ish version of Android then you’ll have something known as Doze mode, basically when the screen is off, Android automatically defers sync jobs so it doesn’t occur constantly and waste battery.