Carry is a role, not an inherent trait of a hero. A carry prioritizes farming during the early stages of the game instead of ganking or pushing. Hard carries take the concept to the extreme, usually avoiding direct engagements with the opposing team for 25+ minutes while they farm up the inventory they need to take over the game.
Because playing as a hard carry puts your team on the defensive for so long it's generally only worth attempting with heroes who scale extremely well with items. Here's a list of the most common options for the role: -Medusa -Spectre -Faceless Void -Phantom Lancer
I'm under the impression that a hard carry is something that is considered a "Right Click Hero" or doesn't do well in the beginning but once they get their items near the end game they will outclass everyone else on the board.