The right hand is responsible for two crucial functions in motorcycling: acceleration and braking. By twisting the grip towards you (so your wrist moves down), you apply throttle (or "gas" the engine.) A little twist goes a long way, so be delicate with this control since sudden engine revs can lead to instability, or even accidental wheelies.
The right hand also controls the front brakes (as seen in the top half of the photo.) Pulling the lever applies the front brakes, and smoothness is crucial here: yank the lever too hard, and the front brakes can lockup, causing the bike to skid, potentially tuck the front end, and crash. Though most bikes are able to stop using a two-fingered technique (as seen in the photo above), some require the entire hand to wrap around the lever. Use whatever technique works best with your bike.
The right foot operates the rear brake (pictured in the lower half of the split-screen photo), but be aware that when the front brakes are in use, the rear becomes less effective (since weight transfers to the front during braking.) Rear brake application is more useful during reduced traction situations and low speed maneuvering, and though rear braking is sometimes more effective on cruisers (because they carry much of their weight over the rear wheel), front brakes are usually the most effective way to stop a motorcycle.