Weight affects the main things wheels do: rotate as the motorcycle moves along, move up and down to follow bumps, twist about the steering axis and leaning over.
Taking the rotation first – before your engine can accelerate the motorcycle, it first has to accelerate the wheel. Lighter things accelerate faster for the same amount of force exerted on them, so cutting mass from the wheels gives a dual benefit to acceleration: these lighter wheels take less force to accelerate rotationally, and since the motorcycle as a whole is lighter, the entire motorcycle also accelerates faster in a linear fashion.
These same rule on acceleration also apply to deceleration (negative acceleration) – or braking. So lighter wheels make braking more efficient too.
The second job is to hold the tire in contact with the road and lighter wheels also make this easier on the suspension. Lighter wheels are deflected more easily (so they react more quickly to bumps), and require less damping as they have less inertia too. As such the suspension can be less stiff, in theory giving better ride and better tire contact. Finally there's the gyroscopic forces generated by two large spinning wheels on each end of the motorcycle. Essentially, the heavier the wheels the less they want to lean or twist. So lighter wheels help you throw the motorcycle around with less effort and move faster when you do.
You only need lighter wheels if you want to brake, accelerate or turn faster, or have your suspension more closely match the bumps in the road. So, any sportsbike could do with them, really. Just remember the rest of the chassis needs to be optimized to suit as well.
Material and production methods are the key for good light-weight alloy motorcycle wheels. Carbon fiber is lighter than magnesium is lighter than aluminum is lighter than steel. No motorcycles have steel wheels now, except stainless steel spokes on wire-spoked rims. Forged is usually considered lighter than cast, although fans of casting suggest that since stiffness is the important thing in a wheel, a cast wheel can be made lighter, since large-diameter hollow-section spokes can only be cast. Generally though, forged metals have fewer structural imperfections, so are stronger, and clever design makes them light and stiff. Forged magnesium is what most MotoGP motorcycles use, which tells you something.