Transnistria exists on territory recognised internationally as part of Moldova. It is divided from the rest of Moldova by the Dniester river (hence then name, though the variation in spelling confuses me). I understand that it is inhabited mainly by ethnic Russians (despite being separated from Russia by Ukraine), while Moldova proper is mainly inhabited by ethnic Moldovans (who may or may not be a subset of ethnic Rumanians). Moldova was part of the Soviet Union, and when the USSR broke up it became an independent state. Transnistria came into being when some local politicians decided that they would rather have their own little country. Russian troops based in the region supported their rejection of Moldovan rule. Since then, Transnistria retains its independence thanks to the ongoing presence of Russian troops. Transnistria also houses most of Moldova's electricity plants, so if the Moldovans ever get bolshy then Transnistria turns off their lights.
As far as I know, Transnistria has no external recognition, although it gets ambiguous support from Russia. Its situation is thus somewhat analogous to that of Abkhazia and South Ossetia. Politically, Transnistria seems to be run by a thuggish authoritarian clique. I understand that the country functions as a kind of Soviet theme park, with its towns still full of red stars and statues of Lenin. I have also read it described as the kind of country you would not want to stay in after dark, particularly if driving in a car with Moldovan registration plates.