The proper way to do netilas yadayim is to hold the cup in the right hand as it is filling up with water, then pass the cup to the left hand, wash the right hand once, then back to right hand,wash the left hand once etc. until both hands have been washed a total of three times. This way applies whether one is right handed or left handed. Some wash each hand four times. It is important to wash both hands alternately (pouring a lot of water at once on each hand is not enough, it must be three times separately) in order to completely remove the ruach hatumah. However if one washed only one hand, the ruach hatumah has been removed from that hand, therefore one who has a cast on one hand should still wash the other hand. If one is in a place without a cup, one should wash their hands under the tap three times, and preferably, open and close the faucet between the washing of each hand. One should be careful that the water used for washing does not spill on the floor where people walk barefoot (side note – if water does  spill, it is not muktzah on Shabbos).