CON is indeed a variety of things as people are saying, just as CHA represents many things (Telling jokes vs. negotiations, people will have different measures of each).
Hit points become tricky as we "fill them up" so frequently. Think of a boxer in the ring, he can take many good blows, but eventually it will take just one (that may or may not be stronger than the others) to knock him out (take him to 0 hp). In between rounds the boxer's corner "heals" him for a partial amount of the damage he has taken, but he won't come out of his corner feeling the exact same way that he did at the beginning of the match, indeed it takes many days of rest to naturally recuperate such wounds without potions (ie medicine) or spells (ie morphine).