shaking hands with a non-mahram woman
mahram-some1 that a women is not allowed to marry e.g father, brother
This is one case where incorrect social customs have taken precedence over the laws of allah, to such an extent that if u try to talk to people and show them evidence that this custom is islamically wrong, they will accuse you of being backward, having a complex, trying to break family ties, doubting their good intentions, and so on.
Shaking hands with female cousins {depends what gender u are, man is not allowed to shake female cousins hands and vice versa}, brothers wives or husbands and uncles wives or husbands, {again, depends on what gender u are, not allowed to shake hands of opposite sexes of the people stated} is now easier than drinking water in our society. However, a careful look at the Islamic texts will highlight the seriousness of this matter.
The Prophet said: "If one of u were to be stabbed in the head with an iron needle it would be better for him than touching a woman whom he is not permitted to touch."
The Prophet said" i do not shake hands with a woman". Aishah said:"No, by allah, the hand of the prophet never touched the hand of a non-mahram woman. He reveived bay'ah{pledge of allegiance} from them by the word only"
It should be notes that placing a barrier sucha as a piece of cloth or gloves is of no use, as the rule of prohibition still applies.