Since 1994 Transdniester has had its own currency, the Transdniester Rouble. Roughly equal to the Moldovan lei in value (€1 = 16MDL $1 = 11MDL) the rouble can only be used within Transdniester and is worthless outside the region. Dollars and even MDL may be accepted within Transdniester but you should change some currency to be on the safe side. You can easily change lei/hrivna/dollars/euros at the train/bus station on arrival in Tiraspol and you can change back whatever you have left over when leaving. Makes a nice souvenir too. Banknotes are issued in denominations of 1, 5, 10, 25, 50, 100, 200 and 500. Transdniester also has its own coinage (kopecks).
Since 1994, Transnistria has been using its own currency. Previously, they had been using Soviet and Russian bills with little stickers stuck on! Today, there are about ten rubles to the US dollar.
Because Transnistra isn't recognized internationally, the Transnistrian ruble isn't of any use outside the region. There are two currency exchanges at the bus and train station in Tiraspol; they will convert your Moldovan lei into Transnistrian rubles, and they will also convert your remaining rubles back into lei when you depart (make sure to buy your bus ticket and snacks first though!). Nobody outside of Transnistria will change your rubles, nor accept them for payment.
I actually kept all of my rubles when I left Tiraspol... after all, it's not often you get your hands on the currency of a country that doesn't exist!