An alkali is a base that is soluble in water. So every alkali is a base, but only the bases that dissolve in water are alkalis. Sodium hydroxide and potassium hydroxide are common alkalis.
When an alkali dissolves in water it forms an alkaline solution containing hydroxide ions, OH-. These solutions have a pH higher than 7.
Copper oxide and magnesium oxide are bases because they react with acids to form a salt and water only, but they are insoluble in water. This means they are bases that are not alkalis. Sodium hydroxide is a base and it is also an alkali, because it dissolves.