The "strength" of an acid or base can be interpreted as its strength as an electrolyte. In other words, STRONG acids and bases are strong electrolytes. As we already know, electrolytes are solutions which conduct electricity because they contain charged particles--ions--and that tells us that like aqueous solutions of salts, solutions of acids and bases contain ions.
Weak acids and bases must therefore not break up into significant numbers of ions in water. By analogy we might compare strong acids and bases to soluble salts and their weak counterparts to insoluble salts. But the comparison is tenuous since acids and bases may dissolve completely in water and still not yield an appreciable number of ions. So there must be additional factors to consider.