Hi nonie! I know it's a little confusing. I hope I can help clarify a bit for you. If you turn off "Use packet data" then yes, you will have no connection to the Internet over any network; AT&T or wifi. It would be like unplugging your computer from your modem or router. The computer will still work, but you won't be able to do anything that involves Internet access like getting mail or surfing the web. If you use Outlook for mail, Outlook can keep checking for mail automatically, but it will never find any because you unplugged your computer from the modem.
Background data just means that applications like gmail, Android Market and the like, can run data transmissions in the background using your Internet connection (Use Packet Data). If you turn off "Use packet data" and leave "Background Data" on, the applications will still try to refresh in the background but they won't get any data because you don't have an Internet connection. Like my Outlook example above.
In short, if you don't want to use any data, turn off "Use Packet Data". It doesn't matter if Background Data is on because it can't work when "Use Packet Data" is turned off.
Now, if you want to "Use Packet Data" (Internet connection) to surf the web or use Google Maps, etc. and you're not on a Wifi network and you also want to minimize the amount of data you use, that's when you'd turn off "Background Data". That way, you have Internet access, but applications on your phone aren't using data you can't see. In that case, you can just manually check for email instead of letting the phone do it for you automatically. Since you're not on an Unlimited data plan, I would leave Background Data off unless you're on a Wifi network. You don't want to forget it's on when you're out shopping and using the AT&T network. Background Data can use a lot of data depending upon which apps you have installed.
When you're home and using your wifi network, you'll need to turn on "Use Packet Data" and just make sure you enable Wifi so your phone will use your Wifi network instead of the AT&T network. You can have Background Data on because you won't be using data from your AT&T plan.
In short, think of your phone the way you think of your computer. If your computer is plugged into the modem, you can access the Internet and Outlook can automatically check for mail every 10 mins.(This is just like the function of Background Data on your phone when gmail checks for new mail in the background). If you unplug your computer from the modem, Outlook will still automatically check for mail, but it will never find any since it needs Internet access to do it.
When you're on Wifi, "Use Packet Data" has to be on for you to access the Internet, but just turn on your Wifi connection so the phone uses Wifi instead of the AT&T network. Here, you would leave Background Data on because you won't be using data from your AT&T plan. You might want to get something like My Data Manager Free from the market so you can track your data usage. There are also widgets on the market that allow you to quickly turn off "Use Packet Data" without having to go into settings.
I hope I didn't confuse you. I know I was a bit redundant, but I hope something here helped a little. If you're still a bit confused about this, ask as many questions as you need to until you feel comfortable with how this works. We're all here to help each other and have a little fun along the way!