A base is any substance that neutralizes acid. It may or may not be soluble in water.
Those bases which are soluble in water are called alkalis.
Thus an alkali is a water-soluble base. For example NaOH, KOH etc.
Alkali, as well as base neutralize acid, but we have observed that alkali produce OH− ions in the water, whereas bases such as copper oxide and ferrous do not produce OH− ions.
This is why it is said that all alkalis are bases but all bases are not alkalis.