What is produced when a base react with water?

If the base is one that contains a hydroxide group (OH-), then the hydroxide base doesn't really "react" with water. The base naturally breaks into the cation and anion, and water molecules surround the different parts.

(Water is polar, meaning it has a positive and negative section. Opposite charges attract, so the water molecules will surround the ions created when the base dissociates, but will not actually react. This polar attraction that water molecules provide also explains why bases acids and ionic compounds will generally dissociate in water).

In the case of certain non OH- containing bases, there is a reaction with water. The classic example of this is ammonia (NH3). When ammonia gas is bubbled through water, the basic gas has an attraction for positive charge, which it can gain by stealing a hydrogen ion (H+, proton) from water (while we write water as H2O, it is better to think of it as H-OH, or hydrogen hydroxide). When the hydrogen is stolen from water, you end up with an OH- anion remaining from the water, and ammonium ions:

NH3 + H-OH <=> NH4+ + OH-

These two charged ions then end up being surrounded by water molecules due to the reasons mentioned in the first paragraph.

Other examples of non-hydroxide bases are all salts from weak acids:

sodium fluoride, potassium carbonate, sodium acetate.

In general:

MeB --> Me+ + B- (dissolving and ionisation in water)

B- + H2O <=> HB + OH- (protolysis in water)