Someone who has strong draconic ancestry, e.g. half their ancestry (one parent, or child of two half-dragons, whatever; someone who was, strictly-speaking, a quarter dragon or eighth dragon might still be modeled with the half-dragon rules). Literally is a dragon, in game terms and in fluff terms, though with a roughly humanoid body shape.
Sometimes ritual or magics could be used to infuse a person with draconic aspects, in game terms becoming a half-dragon. In 3.5e, there was a “prestige class” called the dragon disciple that gained this as its final level, for example. This obviously did not retroactively change the person’s ancestry, just changed their bodies to match those of “born” half-dragons.
In 5e, Hoard of the Dragon Queen is so far the only place where half-dragons have been seen. They are clearly distinct from dragonborn, but the book has actually been criticized by reviewers for not explaining exactly what the distinction is. Based on this twitter exchange...
I expect to learn more of half-dragons in the Dungeon Master’s Guide or Monster Manual.
Originally the “Dragonborn of Bahamut,” these were humanoid devotees of Bahamut, the good dragon god, dedicated to the slaying of the evil dragons aligned with Tiamat. They underwent a ritual to change themselves from whatever race they were originally, into something more dragon-y. The process literally involved getting put in an egg and later “hatching” as a dragonborn.
Despite this process, the dragonborn were still members of their original race, not true dragons. They did lose most of the racial features typical to that race, though, and replaced them with several dragon “aspects,” such as a breath weapon or wings. See Races of the Dragon for more details.
In 4e, these were changed into a more generic draconic-humanoid race, rather than a modification of existing people, and were supposed to be descendants of Io, the dragon greater god.
In 5e, these changed again: apparently in the 5th-edition world, any dragon wishing to bear true dragon children needs the explicit (written?) permission of Bahamut or Tiamat. Failure to obtain said permission results in any children being born as dragonborn, which are humanoids with special abilities depending on their Draconic Ancestry.
In both 4e and 5e, the Dragonlance Campaign Setting changed the interpretation of dragonborn; see Draconians, below.
These are from the Dragonlance Campaign Setting, where they were made by an evil god from the eggs of good dragons; see Canageek’s answer for more details there.
In 5e, Draconians are represented primarily using the Dragonborn race, with a few modifications.
You didn’t ask about these, and they are fairly-likely to never appear in 5e, but just in case you see reference to them and wonder what they are:
This was a template from 3.5e used in cases of someone having fairly-strong draconic ancestry, but not strong enough to use the half-dragon template. They demonstrated their draconic traits less strongly, and in the 3.5e rules, had their original type with the Dragonblooded subtype tacked on, rather than the Dragon type that half-dragons got.
Like I said, I doubt 5e will ever use these, they were primarily required to fill a mechanical niche that was specific to the 3.5e template system.
This is for draconic ancestry even weaker than draconic, used for people where the draconic ancestor is ancient history, or completely forgotten. It has no effect on its own, and dragonblooded people cannot be immediately distinguished from others by inspection. However, they could take a number of feats typically reserved for dragons by virtue of their dragonblooded subtype.
Because the dragonblooded status is a matter of such slight ancestry, it is possible for characters to “discover” such ancestry and tap into it where they previously did not; the Dragontouched feat from Dragon Magic can be taken by any character with at least 11 Charisma, and grants the dragonblooded status (and some other exceedingly minor bonuses). The dragonfire adept class from the same book gains this feat as a bonus feat, and makes no requirements about it; in that case, mere veneration and emulation of dragons is enough to be considered dragonblooded.
From this we can conclude that trace amounts of draconic blood can be found very widely, at least in the 3.5e gameworld.
What the proper definition of a “true dragon” is was the subject of quite a lot of debate in 3.5e, and unlike the above two terms, the term “true dragon” is much more likely to appear in 5e. The metallic (gold, silver, copper, brass, bronze, etc.) and chromatic (red, black, green, blue, white, etc.) dragons are definitely true dragons. Beyond that, it gets murkier. I have made some statements about half-dragons being true dragons and dragonborn, draconians, draconic creatures, and dragonblooded creatures not being true dragons. This I am basing largely on a contentious subject in the 3.5e rules to make an in-character point. You should thus consider these statements with appropriate quantities of salt. Basically, in 3.5e, half-dragons received the Dragon type, while the others all retain their original type.
In 4e, this distinction did not exist within the rules. Player races did not have “types,” and dragonborn were the only playable form of dragon. It is unclear just how dragon-y they were meant to be.
I have not seen official statements in 5e regarding “true dragon” distinctions in that edition, aside from the fact that dragonborn are definitely not true dragons. It is unclear from Hoard of the Dragon Queen whether or not those half-dragons would be considered true dragons.