The Muslim-Croat federation occupying 51% of Bosnia and Herzegovina. The capital city is Sarajevo, other major cities are Mostar, Tuzla, Zenica and Bihać.
This part of Bosnia is governed by Bosnian Muslims and Croats, and is more widely accepted in the international community, as opposed to Republika Srpska. The Federation provides the glimmer of hope that a united Bosnia and Herzegovina could exist once again.
The Bosnian Muslims and the Bosnian Croats (note that the word Bosnian describes locality rather than nationality in this context) are two of the three constituent nations of Bosnia and Herzegovina (the third are Serbs). After starting a war in Croatia in 1991, the Serbian war machine gradually transferred its focus to BiH, so the Croats and Bosniaks (and non-belligerent Bosnian Serbs even!) had to stand together in order to defend from the aggression.
For the two nations (and two religions) to cooperate against a common menace, this was expected and historically consistent. Islam was imported into Bosnia in the middle ages by the Ottoman Turks, who converted the better part of the domestic population, most of which were Croats. (They also converted a lot of Serbs and Vlahs during the same period (these peoples migrated into the western Dinaric Alps because of the Turk conquers in the east), so saying that most of the Bosnian Muslims were once Croats is a point of contention between Serb and Croat historians and of course nationalists.)
Throughout the history of the Croat national movement, the Bosnian Muslims (Bosniaks) were always considered part of the same nation. In fact, they would poetically refer to the Bosniaks as the "flowers of the Croat nation" ("cvijeće hrvatskog naroda"). While this is quite patronizing for Bosniaks who have definitely evolved into a distinct nation of their own, it certainly couldn't indicate that one nation would be inclined to wage war against the other one.
And yet, at one point in the conflict, that is exactly what happened! It became apparent that the nationalists among Bosnian Croats (and to a lesser extent among the Bosniaks) would prefer a complete disintegration of Bosnia based on ethnicity. This coincided with the interests of the Herzegovina profiteer lobby, who made a killing in the smuggling of weapons, drugs, people and whatever else was profitable. Supported by the president of Croatia, Franjo Tuđman, who never really hid his nationalist tendencies, their warmongering would soon overcome the reason and harmony of the past.
In 1992 a series of crimes were committed by mercenaries like Mladen Naletilić "Tuta" and Vinko Martinović "Štela", designed to incite a conflict between the two nations. It is speculated that they were behind an abhorrent terrorist act when a cistern of gas was driven into a Bosniak-inhabited neighbourhood of the town of Travnik and detonated, killing tens of innocent pedestrians.
Tuđman's minion in Herzegovina, Mate Boban, carried out his plans by forming another state within a state and organizing a full scale war against the Bosniaks. It would seem odd that they were not bothered by the fact the two nations were both already at war with the Serbs, were it not for a secret deal that Tuđman and Milošević agreed upon: they would have a pseudo-truce on their fronts during the time it would take them to carve up Bosniak-held territory among themselves!
Bosnian Muslims had no separate "mother country" to back them up like Serbs and Croats did, so opening a new front towards the Croats hit them much more severely. They had to start looking for new allies, and found them in some of the Arabic countries, which then covertly started sending supplies and even mujahedins to Bosnia. Bakir Izetbegović, son of Alija Izetbegović who was the leader of Bosnian Muslims during this time, is said to have travelled throughout the Middle East seeking aid from the Arab countries like Saudi Arabia and Iran.
The new war was particularly bloody due to its surprising, treacherous nature. Previously scattered along the front towards the Serbs, the Croat and Muslim forces started to organize ad hoc to fight each other. Those who moved their units faster would gain an immediate advantage, being able to quickly overpower the other side. This left plenty of room for abuses of the customs of war, and crimes over civilians happened on several occasions. On one particularly sad day, April 16th, 1993, a Bosnian Muslim unit executed 22 Croat civilians and soldiers in a small village of Trusina near Konjic in northern Herzegovina. Later that day, a Bosnian Croat unit ruthlessly retaliated by killing 117 Muslim civilians and soldiers in another village nearby, called Ahmići.
Thankfully, however, in 1993 the fate had twisted: the U.S. Clinton administration finally saw the problem as well and warned Tuđman that he would be treated like Milošević if he persisted in his insane policies of ethnic cleansing. The two sides signed a peace agreement in Washington on March 1st and March 18th, 1994, thus officially forming the Federation of Bosnia and Herzegovina.
The Federation has been created, but peace wasn't instantaneous. Even after the Dayton peace agreement, when the warring with Serbs ended, it still took years to stabilize the structure of the country. The international community installed an Office of the High Representative with the purpose of constant overseeing the progress.
The nationalism was merely supressed, not removed: the Federation still often gets into political crises due to
mostly the nationalist Bosnian Croats, which tend to obstruct federal institutions whenever they deem them not to be in their selfish interests.
That isn't to say there hasn't been any progress; Boban and Tuđman have both died and left room for less militant politicians; Tuta and Štela have both been indicted, prosecuted and convicted for their crimes in central Bosnia by the War Crimes Tribunal at the Hague.
The Federation is still in an economic crisis due to the disastrous effect of the war, but still fares better than Republika Srpska.