How did Descartes' method in his Meditations usher in the era of Modern philosophy?

In the first meditation, Descartes expresses a broad skepticism about knowledge we get from experience. Rather than accepting past opinions of philosophers and thinkers, and rather than accepting that our senses give us the truth of the world, Descartes doubts it all. The only thing he is certain of is that he exists: "I think, therefore I am." 

In the second meditation, Descartes tries to know things in the world in a clear and distinct way, but he must first doubt it all in order to be rigorous in understanding something. So, he uses an example: wax. Since wax can be solid, liquid, different colors, and so on, these qualities are not what make it wax. Thus, he cannot trust his sense experience of looking at wax. He must understand its physical properties: how it melts, why it changes colors, and what makes it "wax" throughout all of these different forms. Therefore, we cannot rely on our sense perceptions of wax to really know what it is. Descartes says we must apply this same method to our "selves."

In the third meditation, Descartes presents his proof of God's existence. Any idea must come from somewhere: ideas we are born with (innate), fictions, and things we learn from experience. Since he has the idea of a perfect being (God), this idea must have come from somewhere or someone. But Descartes is not perfect. Therefore, the idea must have come from God, the perfect being. According to Descartes, he knows this notion of a perfect being because God made it an "innate" idea. Other philosophers will call this kind of knowledge "a priori," which means before experience. These are things we just know intuitively. We are born knowing them or with the ability to discover or remember them.

In the fourth meditation, Descartes discusses human fallibility. We are not perfect, but God is. Even though we have the ability to understand things, "clear and distinct," we also have free will and therefore are subject to error. (The interesting byproduct of this proof is that if God can only make the right, perfect choice, then He doesn't really have free will. This argument can go in circles.)

The fifth meditation addresses God's existence again. In the sixth meditation, Descartes discusses material things in the world. He suggests that if his mind and body are separate, then a thinking thing can exist without a body. Therefore his mind (thinking thing) can perceive/conceive/know a material thing (body). Therefore, material things exist.

Descartes' contributions to skeptical thinking and focus on mathematics would usher the world into Modern philosophy. His philosophical justifications in Meditations would lay the groundwork for rationalism (philosophy using reason) and this put emphasis on the importance of the individual. This shift in understanding the individual in deep philosophical and psychological ways is also a Modern development. Descartes has his critics. In terms of science and philosophy, these would be the empiricists: those who use sense experience to understand the world.