Capsaicin, the chemical that gives peppers their heat, can be effectively neutralized by rubbing your hands in whole milk or a liquid fat such as vegetable oil. After applying the milk or oil to the affected skin, wash with hot, soapy water.
Capsaicin dissolves readily in fat, but water does next to nothing to it. That is why drinking water while you are eating hot peppers does little to stop the pain. Simply washing your hands with water and soap after contact with hot peppers will lessen the burn somewhat, but only because of the washing motions and running water physically carrying the capsaicin away.
Both whole milk and oil are effective remedies because the capsaicin dissolves into the fatty liquid and gets trapped there, in preference to being directly on your skin. Milk is especially effective because it also contains caesin, a protein which binds to capsaicin and neutralizes the pain.
Strong alcohols also dissolve capsaicin, so a high-proof drinking alcohol or a rubbing alcohol meant for first aid is also helpful for relieving pepper burn. Alcohol is less effective than fat, especially for burning in the mouth, but it is a good option for hands if no milk or oil is available.
Finally, acidic liquids, such as lime juice, neutralize the akaline capsaicin; however, they can cause burns of their own when applied directly to sensitive skin.