A person’s hands are considered Tameh ("impure") until he properly washes Netilat Yadayim, which means pouring a Rebi’it (approximately 3.2 ounces) of water on the hand all at once. As such, if a person washed one of his hands, but not the other, the hand that he washed is deemed Tahor ("pure") while the other hand remains Tameh. And therefore if the two hands touch each other at that point, the washing he had done is invalidate, and he must dry his wet hand and then begin the process anew. Of course, he recites a Beracha, as well.
Similarly, if one washes both his hands, and then somebody else, who had not washed Netilat Yadayim, touches one or both of his hands before he dried them, then he must dry his hands and wash anew. In most cases, this happens before the individual recited the Beracha, as the Beracha is recited immediately after drying the hands. Therefore, it is clear that he must recite the Beracha after washing the second time. Some authorities, however, advise the person in this case to touch his shoes before washing the second time in order that the second washing would be required according to all opinions and thus the Beracha can be recited without any doubt. In the rare case where somebody touches one’s hands after Netilat Yadayim while the hands are still wet but after he recited the Beracha, then although he must dry his hands and wash them again, he does not repeat the Beracha.
It must be noted that all this applies if somebody touched a person’s hands while they were still wet. After one dries his hands following Netilat Yadayim, however, his washing does not become invalidated if somebody then touched his hands. During Se’uda Shelishit, for example, it occasionally happens that a person washes his hands and as he walks to his table to eat, somebody greets him and shakes his hand. This is perfectly acceptable and one does not need to wash his hands anew, as the Tum’a cannot transfer from one hand to the next if the hands are dry. This is the ruling of Hacham Ovadia Yosef (Yalkut Yosef – Berachot, p. 201, Halacha 5; listen to audio recording for precise citation), who adds that this applies even if the individual’s hands become wet again and come in contact with somebody else’s hands at that point. As long as they had been dried from the water of the original Netilat Yadayim, they will not become Tameh again if they come in contact with somebody else’s hands.
Summary: If somebody washed Netilat Yadayim and his hands touched somebody else’s hands while they were still wet, he must wash again. If, however, this occurred after he had dried his hands, even after they had again become wet, he does not have to wash his hands anew.
