I have gone to many game conventions where I have seen a "Living" campaign referred to. I do not quite understand what a "Living" campaign is. Explain the overall concept of a "Living" campaign and perhaps how it differs from a "traditional" campaign.
A "Living" campaign is one of the old RPGA-run organized play campaigns designed for you to use and advance a character across multiple play opportunities at public events like RPG conventions and game days, in the same shared world and using officially sanctioned adventures. (For you kids nowadays, the RPGA was a RPG fan organization sponsored by TSR, which again for you kids nowadays is the company that put out D&D prior to Wizards of the Coast.)
Previously, all RPGA campaign play was in competitive format, with pregenerated characters provided. Many of the early AD&D modules were originally RPGA competitive adventures. But they wanted a way to simulate home campaign play a bit more, and so devised a scheme where you could create a character according to more deterministic rules than usual (point buy not rolled stats, which was unusual back then, and set hit points and starting gold). Then as you played approved adventures at events, you'd get XP, gold, and treasure signed off on certificates and/or log sheets by RPGA GMs so that your character could advance.
Living City, set in the Forgotten Realms, was the banner Living campaign from when the idea was initiated (1987) through the end of AD&D 2nd Edition. With the launch of D&D 3e they instead started Living Greyhawk, which was organized in an interesting way - LC was run by a core of RPGA staffers and volunteers, but in Living Greyhawk the in-game countries were parceled out to real world regions and regional coordinators were set up (I was one of the LG Regional Triads for the Yeomanry region). There was also an explosion of smaller Living campaigns, like I played in Living Verge that supported Alternity. There were even a couple supporting non-TSR/WotC games; I also played in Living Spycraft (which of course supported Spycraft). Many of these were small and short-lived. Living games became way more popular than "classic" (pregen) RPGA games, and conventions would often have large mustering areas to form tables (you have to get in at a table with characters of similar level, for an adventure you haven't played yet, which can be complicated - and also a reason why more fringe Living campaigns didn't take off; you really needed a large player base to make table formation at events viable).
In the early days all tracking was done via character logsheets and printed certificates which DMs would sign, to provide traceability. As time went on this went through various incarnations - for example, not tracking individual items but just "gp value" and letting characters buy whatever they wanted (3e days) and trying to track via computer instead (never successful).
World-defining events would sometimes come out of Living campaign play, with the pitch being "send in your tables' results and whatever is most common will become canon!" but this was actually pretty uncommon and was effectively limited to special events at Gen Con and similar - it sounded good but was simply too difficult to administer on a distributed basis (this is all prior to smartphones, ubiquitous wireless, etc.).
With D&D 4e the "Living" campaign branding was retired in favor of Organized Play, and with D&D 5e that has been retired in favor of Adventurers League. Paizo has Organized Play for Pathfinder which works somewhat like Living Greyhawk in that they have regional "Venture-Captains"; not surprising since Erik Mona, now of Paizo, was one of the Circle that ran Living Greyhawk. These have experimented with more computerization, allowing home play of Organized Play events, online play, etc. But they are not technically "Living" campaigns any more, though they do inherit the core idea.
"Living campaign" is a common term for a "shared-universe" campaign played by an extended community of participants, usually mediated by an organization like the RPGA. The idea is that players from all over the world can participate in the evolution of a shared setting, either developing organically based on an aggregate of player actions or pre-determined by metaplot.
How exactly a "living" campaign works has changed over the years: some campaigns of this style may have a standard set of "official" modules for players to play; others might invite groups to have their own adventures as long as they fit into certain constraints (e.g. don't blow up the main setting city on a whim since that might mess up everyone else's fun). In the past, many older ones would assign game-world regions to real-world regions, so that groups who played near each other could conceivably meet to play out their characters running into each other; I think that particular tradition has fallen a bit out of fashion recently.
If you're interested in playing a game like this, a quick google search should turn up a synopsis of the setting and story arc, as well as information about specific requirements to join a game (special rules, org membership requirements, &c.).
There are other shared-world campaigns that don't involve the "living" terminology, such as Mind's Eye Society, a shared-world LARP for World of Darkness.
In a living campaign, your character is imersed into a 'world' with other players you don't know. All activity is posted to a central location, so changes in that world affect everyone who play the game. It also allows you to use one character with different DM/GMs using the same campaign.
Essentially it turns table-top gaming as close as possible to being an mmorpg, except you can actually alter the world. Assassinate a corrupt mayor? Everyone that comes to the town will hear that the mayor mysteriously died.