This is a good question. To say that two countries are interdependent is to say that two countries are related in some sense, usually in a economic way. Let me give you an example.
Currently, there is a credit crisis in Europe. Greece basically is bankrupt and Italy, Spain and Portugal are getting there as well. It might seem that if these countries made bad decisions, then only they should suffer. However, because there is a European Union (EU), other countries will get hurt also. Germany and France are doing well economically, but they too will get hurt because places like Greece and Italy are not doing well. We can say that these nations are interrelated.
We can even say that America is also effected, as America sells their products to Europe. If Europe has no more money to buy, then we will lose out on profits as well. In this sense, we, too are interrelated. In a globalized world more and more countries are all in it together.
We can look at a two-country relationship in the perspective of Ricardo's model of specialization and trade. According to this theory both countries will be better off by specialization and trade. That is, each country specialized in a good that it has a comparative advantage in terms of resources than trade with another country that has advantage in other resources. For example, labor is cheaper in country A and capital is cheaper in country B. Then, country A should concentrate in producing labor intensive products and country B concentrate in producing capital intensive products. By trading these products between these two countries, both of them are better off.
In a simplistic two-country, two-product world we can show that they are better off by specialization and trade. Likewise, this model can be expanded to a multiple-country, multiple-product world and all countries will be better off. This is basis for an open economy whereby all countries are having an interdependent relationship.