You'd want a 6870 + quad + 8GB or equivalent for adequate 1080p gaming. Anything less and you'll need to play on console settings.
thats overkill for 1080p any 5 series or 6 series will not under any circumstance struggle at 1080p obviously some perform better but they will all do it.
a few of the dual core processors will wipe the floor with a quad core becuase it doesn't matter at all if the game doesn't support multicore even a single core will do just as well at that point.
8gb of ram is not required there isn't a game on the market right now that will come close to using 4gb never mind 8.
i use a 4870 and don't struggle on 1920 x 1080 res pushing to 2 monitors and still enjoy decent performance...not great but good enough.
OP if you go for an i3 you will get a better amd for the money but if you go i5 or i7 then you will enjoy a better experience.
in terms of graphics any of the newer ati or nvidia 1gb cards will be more than enough, they all perform about the same in single card setups ati may be slightly ahead but the margins are not really noticeable to you figures on a graph may look pretty when pointing out the difference but you will never notice the difference between 96 fps or 101 fps.
standard ram is 4gb depending on just how much multitasking you do and what they are will change that requirement, photoshop and video editing require more anything to do with rendering will want more if it's there and will help.
the acer all in ones suffer from the same problems as i macs, upgrading is a pain as is repairs they can overheat due to small enclosures and you will be stuck with whatever hardware is there.
a standard tower pc is the better option especially if you like to upgrade.