The full moon on Sunday will be the closest that the moon gets during its full phase this year, so some are calling a "Supermoon." Yes, the moon will appear larger because it's closer to us, but according to Phil Plait, the Bad Astronomer, the effect is not as pronounced as it has been hyped to be. The moon will appear about 1% larger than last month's full moon, or about 10-15% larger than the smallest full moon of the year. Here's the science behind the moon's differing appearance:
As it happens, the Moon orbits the Earth in an ellipse, not a circle, so the distance between us and the Moon changes all the time. When the Moon is closest to Earth in its orbit we call it perigee, and apogee when it’s farthest. These happen once per lunar orbit, of course, about 13 times per year each. This year, the average perigee distance is about 363,000 kilometers (225,000 miles), and the average apogee distance about 405,000 km (251,000 miles) [Note for math and astronomy pedants: astronomers measure distances using the centers of objects, so the distance to the surface of the Moon from the surface of the Earth is a bit smaller than this, by the sum of the radii of the two objects: about 8000 km.]
But those are averages; the actual numbers month by month are all a bit different. The full Moon on June 23 will occur when the Moon is just a hair under 357,000 km (221,300 miles) away, the closest perigee of the year. The phase of the Moon and its distance from Earth are not connected in any way; a full Moon can happen when the Moon is at apogee, perigee, or any point in between. It so happens this June 23 full Moon occurs just 20 minutes after perigee, so it really is about as close as it can get. That’s pretty nifty timing!
So while nothing out of the ordinary will happen during the "Supermoon," you should still go take a look, because the moon is always neat! Link
(Image credit: NASA/Goddard Space Flight Center Scientific Visualization Studio)
As he said, you probably wouldn't notice if it someone didn't tell you and showed a photo contrasting the sizes. As far as subtle effects you probably don't notice about the Moon, the libration of the moon is a lot cooler in my opinion, as illustrated in a (simulated) animation here.