A base is a substance that reacts with acids and neutrlize them. Some bases are soluble in water some are insoluble. Soluble bases are called alkalies.
For example copper oxide does not dissolve in water. Hence it is called as base.
Definition of an alkali: A base that is soluble in water is called an alkali.
In general hydroxides of alkali metals and alkaline earthmetals are considered as alkalies.
Example:
KOH (aq) → K+(aq) + OH- (aq)
Ca(OH)2(aq) → Ca+2 (aq) + OH- (aq)
Therefore, it is said that all alkalis are bases, but all bases are not alkalies.