Things are more stable when the CG is ahead of (a contact point or) the wing of plane. As you move the CG aft, you lose stability and gain 'maneuverability' until you get to a point where the system is no longer stable (there's actually a mathematical explanation which is pretty darn ugly and tedious). A forward CG is good for sport flying for more stability, and also so the nose of the plane pitches down in a stall, thus creating a position to get air flowing over the wings again (i.e. trainers).

3D pilots also benefit from an aft CG in hovering and high alpha maneuvers as the plane is less 'top heavy' and thus easier to hold in a vertical or high alpha position.

Pattern guys do both; some like a forward CG for stability and better tracking through a maneuver, while others like it set a bit further back for a more sensitive plane and less 'push' for inverted flight, but definitely not as much as a 3D plane.

Other things like vertical flight and knife edges are affected by CG too, and usually results in a pitch toward the gear or the canopy. There are lots of trim charts to help adjust for this stuff, but it also involves things like wing incidence, thrust angles, etc. Too much to explain here, but basically things that are designed into the plane for level flight become skewed when you change the plane's attitude.