This is a very broad question and so I'll do my best to supply you with a general definition and then a specific example. You might get an answer closer to what you're looking for by narrowing down the question on time period and geography.

Realpolitik is at its heart the setting aside of "ideological" motivations when crafting political or diplomatic decisions, and instead focusing on materialistic concerns. A diplomatic decision based on ideology would be: "I'm a Catholic King, my neighbor which is a Catholic country is being attacked, I should help defend them." A decision based on realpolitik would be: "I'm a Catholic King, my neighbor which is a Catholic country is being attacked, I should attack them too and take as much of their territory as I can while they are weak."

One excellent example of this comes to us from the Thirty Years War. In 1618 the Catholic emperor of the Holy Roman Empire, Ferdinand the Second, had recently sent two Catholic governors to Bohemia to oversee the administration of the territory. The Bohemians were heavily Protestant and worried that they would be losing their recently acquired religious freedoms, and so they threw the representatives from a window. This is often cited as the origin of the war, and while the war did start along religious lines, with the Bohemians calling upon the Protestant Union for aid and Ferdinand calling upon Catholic Spain to help put down the revolt, as the war progressed the tone and nature of the war fundamentally changed.

When France entered the war, one would expect that the devoutly Catholic country and her chief minister Cardinal Richelieu would enter on the side of the the Holy Roman Empire and Catholic Spain; however, the Cardinal was one of the first masters of realpolitik and often described as a Machiavellian in his political scheming. Spain and the Holy Roman Empire were choking France which was pressed in on by Hapsburg territories. The Cardinal's chief objective was to secure as much power for France as he could, and so started supplying aid to the Protestants. Not only did France encourage, supply, and pay Protestant Sweden to engage in war against Spain and the HRE, but France itself would openly enter into hostilities as well.

If France had based its foreign policy on purely ideological grounds it would not have funded Sweden's war against Spain and the HRE nor would it have openly entered hostilities against Spain and the HRE; however, since the Cardinal based his foreign policy decisions on acquiring as much land, wealth, and power as possible, since he followed a realpolitik approach to foreign policy, he did just that.

The Thirty Years War: The Holy Roman Empire and Europe 1618-48 By Ronald G. Asch