The strength of an acid or alkali depends on how ionised it is in water.

A strong acid or alkali is completely (100%) ionised. For hydrochloric acid

hydrogen chloride (in water) hydrogen ion + chloride ion
HCl(aq) H+(aq) + Cl-(aq)

All of the hydrogen chloride molecules
become hydrogen ions and chloride ions in water
(see examples for other strong acids).

For sodium hydroxide

sodium hydroxide (in water) sodium ion + hydroxide ion
NaOH(aq) Na+(aq) + OH-(aq)

Sodium hydroxide exists as ions both in water and in the solid.
(see examples for other strong alkalis).

A weak acid or alkali is only partly (less than 100%) ionised. For ethanoic acid

ethanoic acid (in water) hydrogen ion + ethanoic ion
CH3CO2H(aq) H+(aq) + CH3CO2-(aq)

Some of the ethanoic acid molecules become ions in water
but most of them stay as molecules.
The reaction is reversible (shown by the arrow).

For ammonia

ammonia + water ammonium ion + hydroxide ion
NH3(g) + H2O(l) NH4+(aq) + OH-(aq)

Some of the ammonia molecules become ions in water
but most of them stay as molecules.