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.
Check out this question: How can I play monsters and NPCs up to their potential?
The challange and XP ratings assume (I assume) that monsters play to their strengths. For example, your Hobgoblin captain should be commanding the hobgoblins - these are not mindless drones who attack without a plan - these are genetically professional soldiers. Good soldiers do not fight fair - level playing fields are for sporting fixtures, not battles.
The hobgoblins should refuse combat unless and until they have the advantage or they are desperate; but these guys plan ahead so they don't get desperate.
If the hobgoblins are aware of the party's approach then they should be preparing a series of ambushes, fall-back positions, escape/disengagement plans, booby traps, ways of separating the party and destroying them in detail.
If they are unaware then they should have done this anyway - good soldiers do.
For example, an attack on the party during a long rest has possibilities. 7 arrows with surprise at the PC on watch should render them unconscious with a high probability of success. They can then move in and slaughter the remaining PCs. Played this way this is certainly a 2x Deadly encounter. If they survive (which is unlikely) they will have earned every XP several times over.
This assumes that the party really wants "near death battles that bring them out bloody and bruised". My guess is that this is not what they really want.
The suggestion of a "battle arena" test is a good one. This can also be done in the campaign - maybe there are gladiatorial contests in your campaign that are not "to the death". Playing a team knock-out (that ends with 1 team knocked-out) could teach both you and your players what "deadly" can really mean without having a TPK.
Best Answer
Entirely different creatures
unitedconquered under a common faith.Volo's Guide to Monsters has some info on this in the section Goblinoids: The Conquering Host:
Volo's Guide later goes more in depth in the nature of their relationships:
Volo's Guide has extensive details about these races that should provide ample background info for worldbuilding purposes.