The first settlers from Terra who came through the Eve Gate built Stargates and colonized various planets throughout the New Eden cluster. When the Eve Gate collapsed, that disaster caused a catastrophic event that destroyed most of their technology and plunged their colonies into a Dark Age. Within a few centuries the majority of those colonies died out.
The few colonies that managed to survive slowly over time regained their lost technology. The first one was the Jove to rise to power. Soon afterwards the Sleepers, Talocan and Yan Jung civilizations all rose to power as well and started rebuilding Stargates to re-colonize New Eden. After a while those 3 died out while the Jove Empire still flourished.
The next civilization to emerge from the Dark Ages was the Amarr who discovered an intact ancient Stargate within their system and successfully reverse engineered that technology. They then constructed the first modern Stargate and started expanding throughout the New Eden cluster. Early on a group of Amarr called the Takmahl rose to great power and then died off as well.
Over the centuries the Amarr expanded through the cluster and eventually encountered the Minmatar who had earlier found and was using a few original Stargates that were still working. The Minmatar had also developed the Acceleration Gate technology which was later copied by the other races. After enslaving the Minmatar, the Amarr then later encountered the Gallente and Caldari and then finally the Jove.
Before meeting the Amarr, the Gallente had already discovered space travel and created their own Stargate. The Gallente introduced that technology to the Caldari who quickly expanded on it, creating their own space warpdrive and Stargate tech. Due to a space race to colonize systems, the Gallente and Caldari entered into a century long war during which the Caldari gained Capsule tech from the Jove. When the Amarr finally met the Gallente and Caldari, they deemed them too powerful and attacked the Jove who quickly defeated them. Soon afterwards the Minmatar rebelled against the Amarr and with the help of the Gallente, gained their freedom.
After that the five Factions came together and jointly founded CONCORD as an interstellar peace-keeping organization. CONCORD brokered peace treaties between the Gallente and Caldari which ended their century long war. Concord also negotiated peace between the Amarr and Minmatar, freeing the Minmatar from threat of invasion and re-enslavement. CONCORD stands for Consolidated Cooperation and Relations Command. It’s main function was to promote trade, improve relations, conduct negotiations and act as a neutral party to mediate any differences between the Factions. Over time Concord increased its own power and influence with funds from collecting customs, confiscating contraband, selling licenses and various other financial means in order to become a Faction independent of Empire control.
Most of the Stargates in unclaimed Null sec were built by the Empires with plans to expand out and claim those regions for themselves. Then the Empyrean Age happened which introduced Capsuleers and due to other various events such as the emergence of Rogue Drone hordes, the Empires give up on colonizing Null sec regions.
Most of the long jump Stargates that circle around the outer edge of Null sec were built by pirates, smugglers, and rebels so they could bypass dangerous routes that lead through Empire space.
That’s basically a condensed version of New Eden Lore. Concord doesn’t own the Stargates, they just oversee them, collect revenue and enforce retribution in high sec for unlawful acts committed by Capsuleers.