Space is a three dimensional area, which makes mapping it difficult without resorting to computer animation or cumbersome angular calculations. Traveller has traditionally solved this issue by ignoring the three dimensional nature of space and presenting maps of Imperial space in the form of two dimensional hex maps in which each hex represents one parsec of space. This has the advantage of rendering space in a convenient and easy to read format at the cost of exact accuracy. For campaign purposes, this format works well enough, so I will keep it. Space is really three dimensional though, even in Traveller, so there may be exceptions to the hex mapping conventions.
The ordinal directions of North, East, South, and West are inapplicable to space, for relatively obvious reasons. Instead, the Third Imperium uses directional terms derived from the relationship to the Orion Arm, the celestial body that encompasses known space. The four directions are Coreward, Rimward, Spinward, and Trailing. Coreward is towards the galactic center, while Rimward is towards the galactic rim. Spinward is in the direction of galactic spin - or clockwise, while Trailing is the opposite, or anticlockwise.
Space is divided by Imperial convention into sectors each measuring 32 by 40 parsecs. Each sector is further subdivided into sixteen subsectors of 8 by 10 parsecs. This mapping convention dates back nearly five thousand years to the early days of the Vilani Empire and has been adopted by all of the Third Imperium's neighbors as an administrative convenience. Other terms used to describe sectors include march, matrix, region, reach, and quadrant. One of the most dynamic regions in the Third Imperium is the Spinward Marches, a sector on the Spinward edge of the Imperium.
The Third Imperium is the largest polity in known space, encompassing nearly 27 sectors of space. Given that a sector holds, on average, between 500 and 650 inhabited worlds, the Imperium commands the loyalty of somewhere between 13,500 and 17,500 inhabited worlds spread out over nearly 35,000 square parsecs of space. These worlds, and many of those in the various client states and adjoining regions have twice been surveyed by the Imperial Interstellar Scout Service, the first survey was published in 420, and the second in 1065. Each survey took nearly one hundred years to complete, and given that the most recent survey was completed fifty five years before the present, this means that the survey data is as much as one hundred and fifty years out of date.
The Imperium is bordered by several client states, minor nations, and unorganized territories, most of which occupy at most a single subsector, and some of which command the loyalty of a single . It is also bordered by six major states. Note that the existence of states dominated by members of particular alien races does not mean all individuals of those species live within their borders. Despite the existence of, for example, the Aslan Hierate, there are Aslan who reside within the Imperium, serve in its armed forces, and even hold titles of nobility. The same holds true of most of the other major races. Conversely, there are also humans who live in polities controlled by aliens, owing loyalty to those nations.
Aslan Hierate: Lying Spinward and Rimward of the Imperium, the Aslan Hierate is ruled over by the Aslan, a superficially lion-like alien race. The Hierate is not so much a single nation as it is a collection of mostly allied clans that share a common culture and a very loose cooperative governing body that has almost no actual formal power. The Hierate spans across 18 sectors. While the Aslan Hierate has never gone to war as a single entity, component clans have allied themselves with various warring parties, and the tradition of sending younger sons out as Iratei to seek lands of their own has frequently caused friction with their neighbors.
Hive Federation: Situated Trailing and Rimward of the Imperium, the Hiver Federation is home to the Hivers, a species of enigmatic and mostly peaceful hexopodally symmetrical aliens that can best be described as starfish with slightly more specialized limbs. The Hive Federation has only been involved in a single war waged against the K'kree, which they won. The Hive Federation controls 17 sectors.
Solomani Confederation: Comprised of humans from Sol living in those territories that were not reintegrated into the Third Imperium following the collapse of the Second Imperium and the coming of the Long Night, the Solomani Confederation is the smallest of the major powers. The Solomani are chauvinistic and nationalistic, having been in multiple wars with the Imperium, most recently the Solomani Rim War in which they lost control of Terra. The Solomani Confederation is directly Rimward of the Third Imperium and rules over 11 sectors.
Two Thousand Worlds: Located to Trailing, the Two Thousand Worlds are the dominion of the K'kree, a six-limbed race of herbivores referred to colloquially as "centaurs". Despite being descended from herbivorous herding forebears, the K'kree are incredibly aggressive, always concerned that meat-eaters they meet will pose a danger to them and not averse to launching preemptive wars to wipe out such threats. Convinced of their own racial superiority, the K'kree treat all other races in their sphere of control as second class citizens and enforce vegetarianism on all within their territory. The Two Thousand Worlds stretch over 14 sectors of space, although they are composed of something more along the lines of 7,000 to 9,000 inhabited worlds.
Vargr Extents: Directly Coreward of the Imperium is the Vargr Extents, populated primarily by the Vargr, a race of bipedal wolves known to be directly descended from genetically modified wolves from Terra. The Extents are not so much a government as they are "where the Varger live", as there is no organization, no central authority, and almost no law. Although the Vargr are disorganized and anarchic to such a degree that it is impossible to say that the Vergr "did" anything as a group, several clans of Vargr allied themselves with the Zhodani and Sword Worlders to form the Outworld Coalition against the Imperium in the most recent three Frontier Wars in the Spinward Marches. The Vargr Extents cover an area of roughly 14 sectors of space.
Zhodani Consulate: This alien dominion of humans is Spinward and Coreward of the Imperium, covering no less than 13 sectors of space. The Zhodani are a major strain of humaniti that achieved Jump capability on their own, and have wholeheartedly embraced the use of psionics throughout their culture and government. The Zhodani have fought five wars against the Third Imperium in the Spinward Marches, the fifth and most recent taking place between 1107 and 1110.