Of the six stats, Constitution represents a character's physical conditioning and heartiness. This means it not only governs how many hit points a character gets, but their defenses against things like poison, disease, and vitality-sapping magic.
Constitution is in a bit of a weird place in the constellation of D&D stats. On the one hand, it's rarely the highest score a character has. It almost never affects, for instance, attack and damage rolls or spellcasting. And it has the fewest associated skills (Concentration is the only one, it's only good for spellcasting and Diamond Mind maneuvers, and that skill was removed in both 5e and Pathfinder outright) of any stat in the game. However, because of its vital role in providing hit points and physical defenses, no powergamer worth their salt wants to make it a dump stat either. Most characters thus get by with a 14 or 16, depending on bonuses and on how many other stats they need.
Constitution is one of the three physical stats for all the reasons above. They only real classes that could be said to depend on it heavily are the 2e paladin and ranger, both of whom had very high ability score requirements (and some "short" races like dwarfs, halflings, and gnomes also enjoyed boosted saves with higher Constitution), the Pathfinder classes that use RAGE (Barbarian, Bloodrager, some archetypes) and scarred witch doctor archetype for the witch, which used Constitution as a spellcasting stat (later changed after beastlike shrieks of rage from countless butthurt wizard fanboys who saw this as honing in on their gig), and the 5e barbarian, who can potentially enjoy the highest AC in the base game via his ability to add his Constitution bonus to his AC whenever he isn't wearing armor, and combo that with a shield. (Genasi and dragonborn also use the stat for some of their racial powers, like innate spellcasting and breath-weapons.)