A strong acid has a conjugate base that is unbelievably weak, so weak that it barely even accepts protons (eg. HCl is a strong acid, but when was the last time you've heard of a solution of NaCl causing the pH of a solution to rise?)
A weak acid has a conjugate base that is weak - but not unbelievably weak. CH3COOH is a weak acid, and CH3COO- is a weak base.
Analogy:
HCl is a married couple with H+ and Cl-. H+ (the guy) and Cl- (the girl) have been married for 20 years. Married in name only. Cl- is not a particularly easy person to get along with, she's quite an unpleasant character. So when you dunk HCl in water, there's a lot of O(2-) ions (other girls) that makes H+'s eyes go O.O. On the down side, there are already two damned guys (H+'s) with each girl - hence why it's H2O. So, at the first opportunity, H+ divorces Cl- and has the time of his life with his...rendevous...with the O(2-)'s, joining the other two H+s to form H3O+
Meanwhile, Cl-, her husband having just left her, tries to find another man. But she finds that no man is interested in her because she's such an unpleasant character; the men would rather be with the O(2-)'s.
In contrast...
CH3COOH is a relatively ok married couple. They have their ups and downs. H+ likes CH3COO-, she's quite nice, but H+ wishes he can have a few on the side. So when CH3COOH enters water...some of the H+ goes "ah, there are some nice looking girls over there each with two men" and then they leave the CH3COO-.
The CH3COO-, being nice enough, manages to attract her ex-husband back, to form CH3COOH again. But the husband cheats on her again and leaves, and comes back again, and leaves...etc. There's a reversible reaction. CH3COOH is a weak acid and CH3COO- is a weak base, but a reasonable base.
Yet in contrast...
H2 is the happiest married couple with H+ adoring the lovely, amazing, beautiful H-. So when you dunk H2 in water...the H+ will see the O(2-) each with two men (to form H2O) and will be like "huh, why do I need those girls when I have my H- babe". So it's impossible to separate these two.
But suppose you dunk H- in water...
When she enters the water, all the H+ who are frolicking with the O(2-)s will turn their heads and stare at her longingly. Before long, they rush away from their O(2-)s and savagely compete for the H-. Eventually, you end up with happily married H2, and some of the O(2-)s only having one man (to form OH-). And that's why H- is a strong base.
A ridiculously unbelievably weak acid like H2 will have a strong conjugate base like H-.
Hope this makes sense...!