Istinja refers to the cleaning of the private parts after urinating or moving the bowels. It is OK to use either toilet paper (istijmaar) or water but using water is preferred.
Always use LEFT hand to wipe or wash.
Using toilet paper: for urine, dab the private part a few times till dry (better to use odd number of times, 3 times is preferred). If after doing istinja and wudoo, you feel that a drop of urine may have come out, it is best to ignore that . It is common for muslims, especially men, to get obsessive about this. After passing stools, wipe 3 times (or, if only using toilet paper due to unavailability of water, wipe until clean).
Using water: it is preferable to use water to wash one's privates well. Again, one should use the left-hand wash off the impurity. Most mosques will have water pitchers or small hand-showers in the rest-rooms.
Three is Enough!
It is common for muslims to become overzealous, even obsessive, with cleaning themselves. This can happen with parts of Wudoo when someone may want to wash the face or hands repeatedly. This is especially common with men getting obsessed about dabbing and squeezing repeatedly to prevent "that last drop of urine." The Prophet (SAW), in his wisdom, advised us "three times" for cleaning our selves during wudoo and istinja so a Muslim's mind can be at peace that the desired level of cleanliness has been acheived.