Is it two-third or two-thirds? If both are correct, how does their usage differ?
A third, like any other unit of measurement, is dependent on the number of units described. For example, we describe 'one apple' and 'two apples'. The same goes for fractional units. 'one third' is simply a single unit of 'third'. Two or more and we use 'thirds'; hence, 'two-thirds' is the correct usage.
Quite correct, but it should be noted that the hyphen is not always used:
The motion was carried by a two-thirds majority.
At least two thirds of business research and development expenditure was found to be wasted.
depending on whether the phrase is used as an attributive adjective (first case) or simply as an adective + noun combination (second).