Entity which occupies 49% of Bosnia and Herzegovina, created by the Dayton peace agreement, and held by Bosnian Serbs. The name "Srpska" means "Serbian", as in, the Serbian Republic. The capital is in the city of Banja Luka, in the western part of Bosnia.
The RS is considered by many Bosnian Serbs to be a state within a state: the Bosnian Serb army ethnically cleansed most of this territory in the 1992-1995 war, making the 1991 census data (when Serb population in the country did not exceed 33%) null and void. The Bosnian Serb political and military leaders of the 1990s like Radovan Karadžić, Ratko Mladić and Biljana Plavšić have all been charged for perpetuating the war against their neighbours at the Hague tribunal.
This part of Bosnia is still known for the lesser human rights for non-Serb people (mainly Bosnian Muslims and Croats), manifested mainly in the Serb refusal to allow non-Serb refugees to return to their homes. Because the population is unwilling to change from their nationalist ways and properly cooperate both with their neighbours and the wider international community, the RS gets cut no slack, receives less investments and more expensive loans, and therefore has worse economic stats than the rest of Bosnia.
Bosnian Croats and Muslims, which are now mostly located in the remaining part of the country ever so unimaginatively named Federation of Bosnia and Herzegovina, feel that any Serb claims on the extra territory are invalid. The Office of the High Representative, which represents the international community, tends to concur with this and the high representatives have so far passed several legal acts in support of the abolishment of the Serb entity.
The actual borders and geographical shape of Republika Srpska were mainly determined based on the Serb conquests in the war, rather than the ethnic maps based on the census. Before the war, Bosnia existed as a single entity because its population was ethnically intermixed, without many ethnically homogenous areas, where one of the three constituent nations would be in a supermajority.
Republika Srpska holds a reasonably wide strip of land along the border of Bosnia and Serbia and Montenegro in the east, with a noticeable intrusion beginning just south of Sarajevo and ending in Goražde (a Bosniak enclave that, unlike Srebrenica, managed to survive the Serb siege during the war). In the west, an area around Banja Luka is also part of RS, and the two chunks are connected with a really thin corridor in the north. The corridor actually splits Federation of Bosnia and Herzegovina into a small, northern part near the river Sava and the larger part to the south.
To understand this better, it's best to look at the maps at http://www.ohr.int/ohr-info/maps/
You may notice that some of those maps consider 66% the lower limit for a predominant majority. For an approach where 85% is the magical number, see http://www.lib.utexas.edu/maps/bosnia/ethnic_majorities_97.jpg
The patches of high contrast in these colorful pictures clearly delineate irregularities in the mapping of ethnic borders to political ones.
It is hoped that the three nations will reconcile one day, and that the two entities will be integrated into a united Bosnia and Herzegovina. Political integration might happen sooner than the real integration -- for people to become tolerant again -- because the wounds of war heal so painfully slowly.