Half-dragon
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.
Dragonborn
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. This has remained their story for 5e.
In both 4e and 5e, the Dragonlance Campaign Setting changed the interpretation of dragonborn; see Draconians, below.
Draconians
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.
Bonus
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:
Draconic creatures
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.
Dragonblooded creatures
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.
True Dragons
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.
Okay, I think I understand the confusion, but I hope I can clear it up for you.
As stated in the sidebar on Chapter 2: wyrmspeakers are all Wearers of Purple. That's the upper-most group in the cult. That same line says that wyrmspeakers can appoint other cultist to preside over cult enclaves, making those appointees also Wearers of Purple.
Going back to the Preface, it says there are no formals grades within the ranks of Wearers of Purple. Presumably that means that to your everyday cultist from initiate up, whether a Wearer of Purple was a dragonclaw (lowest-rank cultist) or a dragonsoul (just under wyrmspeakers), you have to treat them like they were a wyrmspeaker. In other words, Wearers of Purple are essentially the hands and voice of the wyrmspeaker themselves when the wyrmspeakers cannot be physically present.
The Prisoners mission simply says that that mission provides an excellent time to introduce players to the Wearers of Purple through a glimpse of Frulam Mondath, but you could have introduced her on one of the other missions.
Edit: There is also Langdedrosa Cyanwrath somewhere about town. While never explicitly named as a Wearer of Purple, he has enough pull within the cult to release prisoners, his own personal troop of followers, is rumored to be Mondath's right-hand man, and is pictured in purple on page 17. So a DM could have Cyanwrath be the example Wearer of Purple the party first spots. Regardless none should actually be encountered until the end of the chapter, where the party will meet Cyanwrath.
Best Answer
Draconomicon (v 3.5) has the best information I have found to piece together what dragons think like and for the most part they are like geniuses that feel a small urge to eat sheep and hoard treasure.
Are just a few examples from page 25 of the Draconomicon, seek that book for more info.
As for "what does this item make you think like" is somewhat of a different answer as noted in the comments below; the item does not turn you into a dragon with all of the ancestral memories that they would possess. Thus to answer that question "what makes someone dealing with dragons more compatible in mind set?", this would have to give you greater understanding of dragons, more then most; i.e. be closer in alignment to the dragon in question and also have great knowledge on the way said dragon would act or want. The Item does not grant all of these changes but in turn twists your mind so that you simply understand the body language of dragons innately, thus when interacting with them they seem to understand you better and as a result of this understanding you also would understand them to a much higher degree. If said dragon did not know of this item or that you possessed it, it would grant you significant advantage to social encounters with them.
I would describe to my player that while using this item, he suddenly feels everyone around him seems more blunt with their actions, their words seem so plain and open, every non-dragon seems much easier to read and I would grant them a circumstantial buff to their sense motive to any short-lived race. I do not believe they would even notice the item's influence when talking with dragons as it would feel natural.