The Enterprise-D at the distant place
The end of the universe, also known as where none have gone before or the outer rim, was an informal term used by Captain Jean-Luc Picard of the USS Enterprise-D for an unknown area over one billion light years from the Milky Way Galaxy. As of the late 24th century, it stands as the furthest location ever reached by any vessel of any known species.
After leaving the Triangulum Galaxy, which was itself reached due to a mistake on the Traveler's part during a warp engine experiment, another mistake caused the vessel to not only leave Triangulum, but arrive one billion light years away from the Milky Way. However, after the Traveler both explained what he could of their situation and the methods needed for resolving it, the ship was able to return to the same position it had had at the beginning of the experiments.
In the unknown area, unidentified objects of huge size, resembling transparent glowing perfect cubes, sped past in seemingly random patterns. The background was dark blue, possibly a nebula of some kind. The area can cause thought itself to become reality, as experienced by the crew of the Enterprise-D: whatever one thinks actually becomes real, a situation quickly recognized as dangerous for anyone lacking sufficient intellectual control.
Jean-Luc Picard's mother, who came to Picard in a vision, suggested they might be in the universe's beginning, rather than end. (TNG: "Where No One Has Gone Before")
Background
Similar to the description used by Admiral Kirk in 2273 regarding the potential origins of the V'Ger entity, Picard's phrasing of "the end of the universe" vastly underestimates the universe's size. With some quasars having been charted by the beginning of the 21st century at being anywhere between 780 million and 13 billion light years from Earth, the place the Enterprise finally arrived at was most definitely not the end of the universe. In addition, as many theories exist in quantum physics suggesting that space itself is curved, there may in fact be no such place. Other theories, including those of 20th century physicist and Star Trek fan Stephen Hawking, put the edge of the universe at some 15 billion light years away in an area where expanding waves from the Big Bang are constantly forming into new matter. In the end, the phrasing given in the original script is simply anthrocentric in the extreme.