Determining the Relative Price of Trade I'm pretty sound with the concept behind comparative advantage, but still don't get how to calculate the exchange rate in terms of quantities of each products being exchanged, e.g. 1 apple for 2 cars etc.  

Specifically: when two counties, each having two separate comparative advantages in producing two separate goods, wish to exchange these goods, how will the exchange quantity/rate be calculated?
If I understand correctly the question, the trick to calculate the equilibrium (relative) price of trade in the Ricardian model of comparative advantage is to specify demand once the opportunity costs have been determined. Prices of internationally traded goods, like other prices, are determined by supply and demand.

Consider two countries, two goods (computers and textiles), and given unit labor requirements such that in equilibrium the world price of computers will be between 0.5 and 2 textiles per computers. The opportunity costs define the bounds of equilibrium relative prices of trade (0.5 and 2), while the structure of demand determines the equilibrium relative price (p). 

For instance, p=1 can be an equilibrium price such as one country will specialize in computers and the other in textiles. This price will be determined by the interaction of the relative supply and the relative demand of computers and textiles. This interaction is discussed in the Chapter 3 of International Economics: Theory and Policy of Krugman, Obstfeld and Melitz. An interesting video with a concrete example can be found here.