Dimensional analysis is process by which scientists convert between one unit and another using conversion. You were in fact seeing and doing dimensional analysis when we practiced converting temperatures from one scale to another or when we converted ft3 to cm3.
In the following example we actually need to convert two units simultaneously. The important thing to remember when doing this is that the unit you wish to cancel out must be on the opposite side of the fraction. That means if the unit you wish to cancel is in the numerator (top) then the same unit must appear in the denominator(bottom) of the conversion factor.