conjugate acid-base pair
When an acid gives away its proton, it does not simply vanish — it becomes something new that could, in principle, take the proton back. That before-and-after duo is a conjugate acid-base pair: two species that differ by exactly one proton (one H+). The acid is the member with the extra proton; the base is the member without it.
Take HF and F-. HF is the acid; remove one H+ and you are left with F-, its conjugate base. Add the H+ back to F- and you regain HF, so F- is a base and HF is its conjugate acid. Every Bronsted reaction has two such pairs facing off, and the proton hops from one pair to the other: HF + H2O gives F- + H3O+, where the pairs are (HF / F-) and (H3O+ / H2O). A deeply useful rule falls out of this: the stronger an acid, the weaker its conjugate base, and vice versa. Because HCl is a strong acid (eager to lose H+), Cl- is an exceedingly weak base (it has no wish to take H+ back). Because acetic acid is weak, acetate is a noticeably stronger base.
This reciprocity is the engine behind buffers (a weak acid plus its conjugate base resists pH change), behind the quantitative link Ka times Kb equals Kw for any conjugate pair in water, and behind predicting reaction direction: a proton always moves to leave the weaker acid and weaker base, i.e. it settles on the stronger base. Recognizing conjugate pairs turns a tangle of acid-base reactions into a simple ledger of who is more willing to hold the proton.
In HF + H2O gives F- + H3O+, two conjugate pairs appear: HF/F- and H3O+/H2O. Since HF is a fairly weak acid, F- is a non-trivial base; since H3O+ is a strong acid, H2O is a very weak base. The proton mostly stays on HF, so HF is only partly ionized.
Two pairs differing by one proton each; the proton settles where the stronger base wants it.
The conjugate base of a strong acid is essentially a spectator, not a base at all in practice — Cl-, NO3-, and ClO4- do not measurably remove protons from water. Strength is reciprocal: a very strong acid necessarily has a near-useless conjugate base.