Many people have the custom, known as mayim achronim, to pour a little water over their finger tips up to the joints before saying grace after meals. You have probably seen the pretty silver cup and bowl sets that are used for this. Many people, on the other hand, do not have this custom.
This divergence of practice stems from the two principal reasons given for mayim achronim. The first, based on Talmud Berachot 53b, is that we should wash our hands before blessing God to purify them as the priests would wash before their service in the Temple. The second reason is that we should wash our hands after the meal to remove any salt from Sodom, which can be dangerous if it gets in your eyes (based on Talmud Berachot 105b). Tosafot and other Ashkenazi commentators reasoned that since they did not cook with salt from Sodom, it wasn't necessary to be strict about practicising mayim achronim.
The book Ani Tefilati has a nice, non-literal interpretation of salt from Sodom. The people of Sodom were notorious for their lack of hospitality. We wash away any residue of Sodom after the meal, reaffirming our commitment to the needs of our guests and the value of feeding strangers.