Not to touch the impure water on the vessel [or on the other hand]: [1]
One has to be careful that his hands do not touch the water that is on the other hand.[2] Thus one must be careful not to touch the area of the vessel that his wet hands grasped in order to pour on the other hand. [For this reason there are those who are accustomed to have a few handles on the washing cup, in order to avoid this problem.[3] Today our custom is to wash using a towel and thus avoid this issue all together. See Halacha 8 Q&A for the full details of this subject.]
[1] Siddur Seder Netilah; See also Elya Raba 4:4 that due to this suspicion he rules one is to wash in the morning, each hand, three times consecutively.
[2] As this water is impure and the successive pourings cannot remove the impure water of another hand. Meaning if impure water of the right hand got on the left finger, the left hand will remain impure even if he pours on it three times, as the impurity of the right hand cannot be washed off the left hand. [Admur 162:2; Seder Netilas Yadayim 2]
[3] Ketzos Hashulchan 2:14