Chosen in Warhammer Fantasy
The eyes of the Chaos Gods are always on the mortal world, especially the lands around the collapsed Warp Gates through which Chaos enters the world. Any mortal who pleases the Chaos Gods in some way, from killing your father so you can be with your mother and sister before killing them out of boredom or killing every single thing you see for no reason other than the act of killing to donning a pointy hat, can catch the interest of the Ruinous Powers and begin to hear their whispers and feel their blessings. Oftentimes they find themselves lead to or chancing upon powerful enchanted items, gaining followers amongst the wandering warbands devoted to the Four, and before you know it they've mutated into a giant mass of pulp that was once a sapient creature because they had one too many gold stars put next to their name. Most Chosen come from one of the barbarian tribes, most of whom are clad in heavy armor and possess so much muscle that they would make Arnie look bleak.
The more favor they gain and show off their warrior capabilities (being able to slay something really big or prove their tactical capabilities to command), the higher they climb the hierarchical ladder up until they become Chaos Lords themselves, and from there they can either become Daemon Princes, or Chaos Sp...you know wha-wait! No! I didn't even sa-AAAAARRRGLUGRBERGBLESCREEE!!!
Chosen in 40K
A Chosen is a term coined up by Chaos Space Marine Legions/Warbands that describes a particularly powerful member of a Legion/Warband and may have roots in a title attributed to Warmaster Horus Lupercal. A Chosen comes in many forms. They may be very experienced warriors that were chosen by their Chaos Lords to lead squads of Chaos Marines into combat, some are literally Chosen by the Chaos Gods (or a single patron) and given "gifts", others may be powerful Champions that are lesser Chaos Lords serving a more powerful individual...or simply all of those things put together. Whenever the case, all Chosen are characterised by having unholy bling, millennia of hard won experience (stretching back from the days of the Horus Heresy to present days 40K), mutations that are actually useful (this is still up to debate in most situations due to the gods having different moods) and sweet wargear.
Chaos Chosen usually start as normal Chaos Marines that slowly and steadily advance the hierarchical ladder of their Warband. At first the Chosen either lead the less experienced Chaos Marines like their Loyalist kin do, band together under a more experienced Chosen, or even become at one point members of a Chaos Lord's (or any other distinguished Chaos Commander's) retinue, eventually becoming Chaos Lords themselves if they keep up with the villainous heroism/Just as Planned ploys and piousness/satisfying the Chaos Gods/patron god (even going so far as becoming Daemon Princes or those-whose-names-we-will-not-utter if they screw it royally up). Hell, it's actually possible for any Chaos Marine (even a Possessed Marine, a Obliterator/Mutilator, a Hellbrute, you name it) to become one as long as they become better at what they do and show more interest in other forms of warfare.
Interestingly, Chaos Chosen that want to gain Terminator Armour need to gain it in different manners as opposed to their Imperial counterparts. Among Chaos Marines, Terminator Armours are limited (unless they have ties with the Dark Mechanicus, and even then these guys are no different from the Adeptus Mechanicus in term of hoarding stuff, so they rarely give the Spiky Boyz any newly made bling unless they get something valuable, though the Chosen can just directly threaten them too), thus in order for a Chosen to obtain one, they need to take it away from another Chosen either via heavily ritualised combat (aka: the manly duel to the death), or less warrior-like means (good old backstabbing or "a series of unfortunate accidents"). Krumping a loyalist Terminator and managing to salvage his armor is also one of the available options open to them.
Additionally, Chosen, as mentioned before, have either the favour of all the Chaos Gods, or a single Patron. Thus there are Chosen that are either Undivided, or serve a single Chaos God. Mixes of single god worshipers among Undivided Warbands are as common as those who worship Chaos as a pantheon, and each god gives unique Chaos Gifts to its Chosen. Additionally, Chosen also gain wargear that befits someone that either serves Chaos Undivided or a single patron god.
Aside from a single run of figures way back in the Dark Vengeance box set, these guys have never and quite possible will never get a plastic kit. Kitbash solutions vary from beefed-up ubermarines to Chaos special forces - now 8th edition went and made them the backbone of most competitive lists, you might as well get to work on your own.