Aquinas thinks that part of what is distinctive about rational creatures like human beings is that they can act for an "end" that they choose to pursue. (I.1-2) "End" is a technical term in Aristotelian philosophy, which just refers to one's purpose or goal. In other words, it makes sense to ask a rational creature, "Why did you do that?" and to expect an answer of the form, "I did it in order to achieve such-and-such a goal." Aquinas admits that humans sometimes do something without any conscious end, like absent-mindedly playing with your hair while you talk. But he says that you are not acting characteristically like a human when you do things like that. (I.1, Objection and Reply 3) One way of understanding what Aquinas is saying is this: If you went through life without choosing ends, you would be acting like a zombie or an object rather than like a human being.
Aquinas also thinks that you choose an "end" only if you regard it as good in some respect. The goodness of an end does not have to be what we might call "moral goodness." The end you pursue just has to be something that is "good" in some broad sense. In order to understand why this is a plausible claim, consider the following example. Suppose we knew someone who was collecting lots of pins. We asked him what he wanted all the pins for, and he said, "I just want pins." We ask him if he plans to use the pins for something else, like sewing. He says, No. We ask him if he is a collector (the way some people collect baseball cards or beenie babies), and he says, "No. I just want pins." I think we would say that, if it is even possible for there to be a person like this, he would be fundamentally irrational, unless he could explain to us what was good about having lots of pins.
Among your ends, you want some things only as a means to something else. For example, if you have a job that you don't particularly enjoy, and I ask you why you stay at it, you might say, "I need to work in order to make money for food, rent, books, and stuff like that." So working, in this case, is only a means to some further end. Is it possible that everything you do is like that? In other words, is it possible that every end you have, you seek only as a means to some further end? Aquinas says that it is not possible. Aquinas thinks that there must be some ends that you choose for their own sake, and not for the sake of any further ends. Why? The whole point of an "end" is that it explains why you do something intentionally. Now suppose that everything you do intentionally you do for the sake of something else. Then there would be an infinite series of things that you are pursuing. ("I do A for the sake of B, but B for the sake of C, and C for the sake of ....") But then there would never be any real explanation for why you do something intentionally, because we could never point to anything and say, "Okay. This is what she is trying to get through her actions." (I.4)
Something that you want for its own sake, Aquinas calls an "ultimate end." How many ultimate ends do you have? You might think that you can have several, but Aquinas will argue that, whether you know it or not, you can really only have one ultimate end. Aquinas gives several arguments for this conclusion, but the argument you will probably find easiest to understand is the following. Suppose you think you have two ultimate ends: pleasure and virtue. Since you think these are ultimate ends, you want each for its own sake. Now, Aquinas will argue that neither one of these, by itself, could be an ultimate end. Because if you had only pleasure, you would want something else: virtue. But then pleasure and virtue is a more ultimate end for you than pleasure by itself. So you don't really have two ultimate ends, pleasure and virtue individually; instead, you have one ultimate end, pleasure and virtue together. Another way of putting the same point is that pleasure and virtue are not distinct ultimate ends, but are rather constituents of your one ultimate end. (I.5) (Note that Aquinas is here assuming that any two things you could genuinely want for their own sake individually could be combined into a greater end. Are there are any counterexamples to this claim?)
So Aquinas thinks he has proven that each person has one ultimate end. He next argues that all humans, whether they realize it or not, have the same ultimate end. He calls this ultimate end beatitudo, which literally means "blessedness," but is rendered in your translation as "happiness." The translation of this particular term seems unsatisfactory to me, because the English "happiness" is more specific than the Latin beatitudo. We tend to think of happiness as a pleasant psychological state. However, beatitudo is more general: it is whatever it is that is the ultimate end for humans. So, in a way, the statement that humans seek beatitudo is, by itself, almost trivially true. (Saying that humans seek beatitudo is like saying, "Everyone wants to live well," where "living well" does not necessarily mean living morally. Who would disagree with this?) Nonetheless, although all humans agree in wanting beatitudo, they disagree about what beatitudo consists in. (I.8, V.8) (This is why "happiness" is a little misleading as a translation of beatitudo, because "a pleasant psychological state" is one specific answer to the question, "What is beatitudo?") But Aquinas' argument in the Summa is by no means trivial, because he tries to prove that there is really only one thing that beatitudo could be for humans. In other words, people in life pursue many different things -- wealth, fame, power, sex -- believing that possessing that thing will give them the best life (what we might call "a charmed life"). But there is really only one goal that would give you the best life if you get it.
What Aquinas does in Questions II and III is to try to show that certain things people commonly pursue cannot be the highest good, and something else (that some people do pursue, and everyone should pursue) is the highest good. Question II, Article 1 tries to convince you that possessing wealth is not what makes you "happy." In Articles 2 and 3, he tries to convince you that neither honor nor fame will make you happy. In Article 4, he argues that happiness is not just possessing power. In the 5th and 6th Articles, he argues that happiness is neither having any good of the body nor is it physical pleasure. Article 7 is a little tricky. He says there that "happiness does not consist in any good of the soul" (II.7) However, Aquinas clarifies what he means by this by saying that happiness "is a good inhering in the soul. But that ... which makes man happy, is something outside the soul" (II.7, Reply 3). In other words, your soul is what is happy, but it is made happy by something outside of it. Finally, in Article 8, Aquinas tries to convince you that there is nothing created (i.e., nothing in this world) that can give you happiness.
So what can give you happiness? Aquinas answers this in Question III, "What Is Happiness." In Article 5, Aquinas discusses whether happiness is an activity of the "speculative or practical intellect." In other words, is happiness using your mind to contemplate theoretical truths, or is happiness using your mind to figure out how to solve practical problems? Aquinas argues that it must be the former. In Article 8, "Does the Happiness of Man Consist in the Vision of the Divine Essence?" Aquinas argues that happiness consists in a certain kind of theoretical knowledge: the experience of God that we achieve after death (if we go to Heaven).
This structure might seem forced and artificial to you, but Aquinas uses it for several reasons. He wants to make sure that he has given serious consideration to every objection to his own view, and that he has clearly stated his argument for believing in his own position, and that he has clearly stated his response to the major objections to his own position. However, sometimes it is easier to understand an article if you read the parts in the following order:
Let's apply what we've learned. Go to pp. 15-16 in your translation, and take a look at II.1. As we saw, the general topic of this question in the Summa is, "In What Man's Happiness Consists," and the specific issue that this article examines is "Does Man's Happiness Consist in Wealth?" We go to the "On the contrary" section, and see that Aquinas' view is that "...happiness does not consist in wealth." He cites Boethius, an earlier philosopher, as an authority on this point.
Next, we read Aquinas' argument for this conclusion in the "Response." Basically, Aquinas argues that wealth is only good as a means to other things, and is not worth pursuing in itself, so it cannot be an "ultimate end." More specifically, Aquinas begins by distinguishing two kinds of wealth: "natural wealth," like "food, drink, clothing," and "artificial wealth," "such as money." He argues that human happiness cannot consist in articifical wealth, because we want money, for example, only to buy other things like food and drink. And human happiness cannot consist in natural wealth either, because you want things like food and drink, not for their own sake, but only "for human sustenance." In other words, you want food only in order to satisfy the needs of your body. So even if you need food in order to be happy, having food cannot, by itself, be human happiness, because all the value food has depends on something more important: the health of your body. (I know what you are thinking now: "But I also want food and drink because of the pleasure they give me. Isn't bodily pleasure happiness?" Aquinas considers this possibility in II.6 [Question II, Article 6]. You may not agree with Aquinas, but almost every possibility and objection you can think of is dealt with by Aquinas somewhere in the Summa.)
Now you should go back and read each objection to Aquinas' position, followed by the corresponding "Reply." I'll walk you through the first Objection and Reply, and leave the second and third pairs for you to think through on your own. The first Objection basically argues that wealth "has the greatest hold on [our] affections" (indeed, Aquinas notes that a Bible passage seems to say this!), so wealth must be what our happiness consists in. In his Reply, Aquinas admits that there are some people "who are conscious only of goods which can be acquired with money," but suggests that such people are "foolish," and that "our judgment about human goods should be taken, not from the foolish but from the wise, just as we should take our judgments about taste from those with a well-ordered sense of taste." This reply is challenging to our 20th-century perspectives, because we often assume that there are not experts or "wise" people in matters of either taste or human goods. But this is precisely what Aquinas questions. I think Aquinas would also say that, in his Response in this Article (and in other Articles in this Question), he has established by argument that goods acquired by money cannot be the only goods.