CAVEAT: I can't claim to know too much about Abkhazia, so I am willing to take corrections from my many readers on any factual inaccuracies contained here.
Abkhazia is a separatist region of Georgia, the country in the Caucausus that used to be part of the Union of Soviet Socialist Republics. In the past, the region had a degree of autonomy within Georgia, but when the Soviet Union broke up a separatist movement came into being. Perhaps the Abkhaz separatists feared that their distinct ethnic identity would be swamped in an independent Georgia, or perhaps there were more sinister forces at work. Either way, the Abkhaz separatists successfully fought off Georgian armed forces and established a de facto regime in the former autonomous region. This achievement is all the more impressive when one recalls that ethnic Abkhazians were apparently only a minority of people in the Abkhaz autonomous region.
Since the war (which took place at some point in the 1990s), Russian troops have been deployed in Abkhazia, supposedly as peacekeepers between the separatists and the Georgians. It is widely believed, however, that the Russian troops are primarily there to protect the separatist regime and prevent the re-absorption of Abkhazia into Georgia. There are even those who see the whole business of Abkhaz separatism as a scheme of the Kremlin to weaken Georgia and undermine its independence, a proposition supported by the astonishingly well-armed and trained forces the Abkhaz separatists were able to deploy against the Georgian state. Further evidence of Russian partiality was seen recently when the Georgian flew an unmanned drone over the separatist region, only for it to be shot down by an unidentified jet. The Abkhazians do have their own air force (largely consisting of First World War biplanes and balsa wood aircraft powered by rubber bands), but the unidentified jet had the kind of twin tail-fin only seen in the latest Russian air force interceptors.
I am not clear on whether the Abkhaz separatists wish to set up their enclave as a little independent state, or whether they would ultimately prefer to merge it into Russia. Given the apparent links between Abkhaz separatism and the Russian state, it is perhaps not really appropriate to think in terms of the separatists as having any actual autonomous goals and desires – they may well be simply creatures of the Kremlin, people whose goals are defined by Russian political interest.
Picture from Wikipedia