Heres a little breakdown for you:
Background data is well background data
This allows the system to use data in the background and in the Android market. i.e if you disable this you can NOT activate auto-sync either. This is required to be on to use the Android market {browsing and downloading apps from it}.
Auto-Sync is the ability to :
1) Automatically sync your contacts to your Google account {I would suggest leaving this on as if you lose your phone your numbers are safely backed up in the cloud and are easily accessible on the web}
2) Auto-Sync your calender so any changes you make to the calender on your phone (or pc) will automatically sync together so they are both up to date {PC & on your mobile}. e.g if you update your calender on the phone it will automatically update it online. Same as for the contacts
3) the ability to receive Your googlemail automatically. You know if you disable this one your googlemail will not be pushed to your phone. You will have to manually refresh your inbox.
So I would suggest leaving them all on unless you have to turn them off for any reason. If you disable them you obviously effect the ability of your phone until you turn them back on.
Hope this answers your question
Carl