What are the different approaches to the study of political science?

Political science studies political processes, institutions, and behavior to attempt to understand the political environment both in the present and historically. An approach to political science should be capable of describing what exists, explaining why it exists, and predicting what might exist in the future. Historically, however not all approaches have always done all three of these things.
For instance, Plato's approach to political science is called a Normative approach. Plato skipped the describing and explaining steps in order to spend his energy dreaming of what he thought a perfect world would be like. Many other political philosophers, including Rousseau and More, also used this approach.
As the desire to describe and explain emerged, approaches which relied on analyzing the state, its institutions, and its laws became popular. These approaches consider past and present to analyze what has been in order to determine what could be. These approaches are considered Traditional ways of looking at political science.
These traditional approaches had gaps as well, however, and the modern age has brought more kinds of questioning into the mix, by expanding its study to include more than just political institutions themselves. Modern approaches seek to understand the people who participate with the political system and the environments they live in in order to see how these things affect the system as a whole.

In order to properly understand the different approaches to the study of political science, we have to first break the topic down into two general categories: modern and traditional. And then each of those has its own subcategories.
Let's first explain the difference between modern and traditional. Then we'll break these down into more specific categories.
Traditional approaches to political science focus on the study of the government. How does the government function? How is it organized? What would an ideal state look like? These are questions that this approach would address. This traditional approach is idealistic and more concerned with history than science.
There are four kinds of traditional approaches:

Philosophical. This is the oldest approach, with roots in the philosophies of Plato and Aristotle. It's idealistic and concerned with ethics, with what is good and bad in a state or society. This approach focuses on the establishment of an ideal government with a standard set of values and social norms.

Historical. As the name suggests, this approach is based on the idea that we can only understand the present political situation by considering the past. Machiavelli is a famous proponent of this way of thinking, as he believes that history and politics are intricately and inextricably linked.

Legal. This approach is based on the idea that the government is meant to create and enforce laws. The legal process and how justice is served are key concerns of this approach. Examples of thinkers who embodied this approach include Cicero and Jeremy Bentham.

Institutional. This approach is concerned with the formal branches and structures of government: political parties, legislature, the judicial system, and so on. In this approach, the organization of political institutions is of critical importance. Like the philosophical approach, it has roots in the philosophy and teachings of Aristotle.
Now, onto modern approaches.
Modern approaches to political science are, in part, a reaction to the traditional approaches discussed above. Contemporary thinkers introduced these approaches to fill in the gaps left by the traditional approaches in use for so many years.
What are those gaps? Well, traditional approaches are focused on history and formal structures. Modern approaches, in contrast, embrace a multidisciplinary way of thinking, and they embrace science more than history. They're not just focused on empirical data, but on what we can extrapolate from that data.
There are two key subcategories under the general category of modern approaches:

Behavioralism. Initiated by David Easton, this approach is all about predicting what will happen based on what has already happened (i.e., using empirical data to draw conclusions, as I mentioned above.) The idea is that we can observe and make generalizations about political behavior and that those generalizations can help us figure out what to expect in future political situations.

Post-Behavioralism. Modern approaches are a reaction to traditional approaches, and post-behavioralism, in turn, is a reaction and a reform (and, in some ways, a continuation) to behavioralism. The post-behavorialist approach is based on the notion that behavioralists were too focused on methods and techniques and that not every problem can be neatly solved. Not every future situation can be predicted by generalizations and stereotypes, post-behavioralists argue.