Money that you save for the future is a nest egg. Your parents might be saving their nest egg so they can move to Mexico when they retire.
A nest egg is usually a retirement account, a fund you add to regularly so your savings grows every year. You can really describe any cache of saved money as a nest egg — even a piggy bank, growing gradually more full of dimes, is a nest egg of sorts. The term has been around since about 1700, meaning "retirement savings." It comes from the practice of leaving a real or fake egg in a nest to encourage hens to lay.