As we know, a weak acid ionizes just “partially” in water ( In fact the essential difference between a week acid and a strong acid is that whereas a strong acid dissociates or ionizes to the full extent in an aqueous solution, a week acid does not).
Because of this incomplete/low level of ionization, there remains chemical equilibrium in any weak acid-and-water solution (and FYI, this is the case with the ionization of weak bases also).
If the weak acid is "A", the dissociation reaction can be represented as follows:
AH + H2O ↔ H3O+ + A-
Now, this is also called as protonation of water; the acid donates a proton/H+.
Let us take the example of Acetic Acid, which is a typically a weak acid.
Acetic acid dissociates as follows:
CH3COOH = CH3COO- + H+
The ionization of Acetic acid in water takes place as follows:
CH3COOH + H2O ↔ H3O+ + CH3COO-
Acetic acid Water Hydronium ions Acetate ions
(Aqueous) (Liquid state) (Aqueous) (Aqueous)
"↔" represents that this reaction is in the state of Chemical Equilibrium.
And further to this, the equilibrium constant for any weak acid ionization in water (which now we understand is not 100%) is called as acid dissociation constant and can be represented as Ka.