In this reaction:
Ammonia ([tex]\(NH_3\)[/tex]) is a weak base. Ammonia is a weak base because it only partially ionizes in water. When dissolved in water, ammonia accepts a proton (H⁺) from water to form ammonium ion ([tex]\(NH_4^+\)[/tex]) and hydroxide ion (OH⁻), but this reaction does not go to completion, meaning a significant amount of ammonia remains un-ionized in the solution.
[tex]\[
NH_3 + H_2O \rightleftharpoons NH_4^+ + OH^-
\][/tex]
Ammonium hydroxide ([tex]\(NH_4OH\)[/tex]) is a weak base solution. Ammonium hydroxide is the name often given to an aqueous solution of ammonia. In water, it exists mainly as ammonia ([tex]\(NH_3\)[/tex]) and water and has only a relatively low concentration of ammonium ([tex]\(NH_4^+\)[/tex]) and hydroxide ([tex]\(OH^-\)[/tex]) ions, similar to the reaction shown above. Therefore, it is also considered a weak base. The term ammonium hydroxide is just a conventional way of referring to an aqueous solution of ammonia, where [tex]\(NH_4OH\)[/tex] denotes the dissolved ammonia in water, which acts similarly to a weak base.
