finite sets are trivially compact, if you take an open cover each point is covered by definition, just take one element of the cover for each point and it is finite since there are a finite number of points.
The converse is just as trivial if you attack it the right way. X has the discrete topology and is compact. Suppose it were infinite for a contradiction. Well each point in X is open under the discrete topology, so take this as your open cover. Try to find a finite subcover of this. Clearly not possible since if you removed one of these open sets from this cover it no longer would cover X thus X cannot be infinite and therefore must be finite.
Hope that helped