Ironborn
"Ironborn is a cooler way to say "Self Found." It refers to items or characters or a play style that uses only items a player finds themselves. There are many different gradients of being Ironborn, and different players are more or less "purist" in their approach. Some players will use only items, gold materials, etc that their character finds personally. Other players allow sharing of items and gold across an account (this is hard *not* to do in Diablo 3 with the stash and shared Artisans and gold), while others will trade or sell items out, but will only use gear they find themselves.
Ironborn grew in popularity during Diablo 3, as many players felt the Auction House was corrupting and an impure way to play, since it made buying and selling of gear too easy.
Ironically, D3v2 and then especially Reaper of Souls came along and turned everyone Ironborn, with the extensive item binding in RoS locking every legendary, enchanted item, gem, material, gold, and more to the account that found it. These game changes essentially killed the Ironborn movement, even bringing an end to Waterfiend's popular Ironborn column on Diablo.IncGamers.com.
Once everyone was Ironborn by default, there wasn't anything to write about as a special purist play style choice.
Ironborn Mode
Before Reaper of Souls came and enforced self-finding as a play type, many players were hopeful that Blizzard would offer some support for the Ironborn play style. Such as by giving players a way to designate themselves as Ironborn, locking them out of using the Auction House or trading gear. Obviously anyone could do that on their own, but the point was to create a community of players following the same rules.
The larger theory was that the game could actually include an Ironborn mode which would function much like Hardcore does. Players would designate their characters as Ironborn upon creation, (they could be Hardcore as well if they wished) and Ironborn characters could never use the Auction House and could only play games with other Ironborn characters. That's essential, as the whole point is to create a "clean" economy and a level playing field.
The Ironborn concept is a reaction against the great ease with which high quality gear can be obtained via the Auction House and the lack of a single player mode in Diablo III which would let players stay entirely self-contained, or play only with their circle of friends via LAN or TCP/IP.
Many players thought that would never happen since Diablo 3's long term profit model was taking a cut of the real currency sales from the RMAH. That, at least, was proven wrong by the shutdown of the entire Auction House in early 2014.
Background of the Term
The term was coined and popularized by Flux, webmaster of Diablo.IncGamers.com, in late 2012. It's inspired by the Iron Islanders in George R. R. Martin's Song of Ice and Fire fantasy series. In Martin's series, the Ironborn are fierce warriors who feel all jewels and finery must be taken by force, "paying the iron price" rather than bought "paying the gold price," as the softer men of the mainland do.