This is what I saw from Chemistry: The Central Science 13th ed. by Theodore brown, page 698. It stated that there are two types of weak bases: one is those with lone pairs of electron and the other is anions of weak acids, which means conjugate bases of weak acids. My question is, aren't conjugate bases of weak acids strong bases? Because weak acids are less readily to give off proton but it's conjugate base can accept proton easily.
Your understanding is correct - on a relative scale, the conjugate base of a weak acid is a strong base, and the conjugate base of a strong acid is a weak base. For example, hydroxide ion (conjugate base of water) is a stronger base than acetate ion (conjugate base of acetic acid) which is a stronger base than chloride ion (conjugate base of hydrochloric acid).