The excerpts (from Aquinas - Political Writings, ed. Dyson) you'll be reading by Aquinas are all from his great work, the Summa theologiae. Aquinas writes in a very predictable and rigorous form. The basic unit of text is called an 'article' (in latin - 'articulus', plural 'articuli'). Each article is organized as follows:

 1) The Article Topic - A Question

    Each article topic begins with a question on a certain position. For example, on page 72, we find  articulus 2:

"Whether Christians are bound to obey the secular powers, and tyrants in particular."

2) An Answer - The Position Aquinas will be arguing against

    Immediately following the question of the article Aquinas states an answer. But, this is important, it is not the answer that Aquinas himself holds. It is the position that Aquinas will be arguing against. At first, this might seem confusing, but it's actually rather helpful. Since you know that Aquinas will be arguing against this statement, you can read the rest of the article in light of that knowledge.

For example, on page 72, the initial answer is the following:

"It seems that Christians are not bound to obey the secular powers, and tyrants in particular."

So, you know right up front that Aquinas believes that Christians do have to obey secular powers, even tyrannical powers.

3) The Objections, or, Obiectiones

Immediately following the position Aquinas will be arguing against are what are called obiectiones, which just means 'objections'. Now, be careful. They are objections in the sense of being objections to the final position that Aquinas will hold, so they are actually in support of the answer to the question above them. So, part (3) is in support of part (2)

4) Sed Contra ('On the Contrary')

After the objections is a short statement of Aquinas position, in favor of an answer to the initial question (1) that is the negation of the answer given in (2).

5) Responsio ('Responses', sometimes, instead, Solutio or 'Solution')

This is a longer statement of Aquinas overall response to the initial question. 

6) Specific Answers to Specific Objections

Aquinas then finishes up by making arguments against the specific objections to his view given in part (3).

A Couple Of Other Points:

By the phrase "The Philosopher" Aquinas means Aristotle.

By the phrase "The Apostle" Aquinas means the apostle Paul of Tarsus.

Aquinas makes many Biblical references. A good source online to help see some of these passages in context is the standard King James version.  

- That's it. It can be a little difficult at first, but after you get used to it the form is very clear. For more on the organization of the Summa, read Dyson's introduction, especially around page xxii.