Giving a baby water can be very dangerous if he or she is under 6 months of age. Many of us probably already know this, but it's worth posting.
According to the September 2008 issue of Parents, "[You shouldn't] give extra water to infants younger than 6 months and don't give more than six ounces a day to babies between 6 and 12 months of age." Water causes a baby's kidneys to flush out sodium, which can cause a seizure. "The best way to hydrate a thirsty baby is with breast milk or formula-- and you should never dilute formula with extra water."
I have heard of pediatricians recommending water or slightly diluted formula to parents of chronically constipated babies, but how much depends on your baby's size. Obviously, never do this without consulting your child's pediatrician first.
For more information on babies, check out www.parents.com/common/magazine.