Aristotle's little nugget, dry and semi-opaque as it may appear, is a conceptual cornerstone of western thought and is deserving of careful consideration by anyone seeking a happier, more fulfilling life.
To begin, some linguistic housekeeping:
1) Aristotle's word found here as "happiness" is the Greek eudaimonia. I don't know Greek, but most sources on the subject take pains to tell us that "happiness," though a common English translation of eudaimonia, is insufficient. "Human flourishing" is said to be a better rendering. However translated, eudaimonia was considered by Aristotle to be the answer to the question "What is the aim of human life?"
2) What about "soul"? Simply put, for Aristotle, soul is the life principle. Things that are alive have this "life stuff," everything else doesn't.
3) "Virtue" is a translation of the Greek arete. Virtue is a fine word, but arete can also be helpfully understood as "excellence." Specific to the present context, virtue can be read as "excellence at being human."
With this understanding, let's try a clarifying revision of Aristotle's thought: A happy life—a flourishing life—is one lived in pursuit of virtue, i.e., human excellence.
So far so good. The next move, and you'll have seen it coming, is to define human excellence. What does it mean to excel at being human? It is to nurture and make good use of those capacities that are uniquely ours as human beings. Here is Aristotle's reasoning: Things in this world have specific functions—knives cut, plants bear fruit, homes protect us from the elements, etc. We judge a thing as excellent when it perform its particular function well. An excellent knife cuts easily and reliably, an excellent plant bears high-quality fruit, an excellent house is sturdy and comfortable and provides good shelter.
As human beings, we, too, have a function that is specifically our own: We can reason. We share a lot with the beasts—we are physical creatures, with all the biological operations and limitations that implies—but mankind is also endowed with the power of abstract thought and higher reason. Since reason is our uniquely human function, then, just as an excellent knife is one that cuts well, an excellent human is one who reasons well. And the highest aim of man (and the source of our happiness and flourishing) is to live a life in accord with reason.
Next step: What does living in accord with reason entail? As a species, we apply our reasoning power to all sorts of things, from choosing our breakfast to solving obscure problems of quantum physics to fulfilling our responsibilities as friends, neighbors, and members of society. Aristotle proposed that this broad variety of thoughts, decisions, and activities that make up our day-to-day lives can be categorized based on the types of excellence (virtue) at which they aim and that, fundamentally, they can all be slotted into one of two main categories: intellectual virtue and moral virtue.
Among the intellectual virtues, Aristotle named scientific (empirical) knowledge, wisdom (both theoretical and practical), and craft know-how (such as skill employed in the arts). Our rationality gives us the capacity to develop these kinds of virtues, and they are imparted to us mainly through instruction.
The moral virtues are concerned with practical living—getting along in the world—and include courage, friendliness, justice, prudence, self-discipline, and others. We are not born with these virtues, but, as Aristotle says, "we are adapted by nature to receive them" (i.e., our rational nature helps us recognize their wisdom and utility), and they develop in us only by conscious habit. We must foster them.
Happiness, then, consists in steering our lives toward intellectual and moral excellence. (It's not a question of having to attain perfection; the fundamental orientation is the important thing.) Our minds are powerful, and we will find our fulfillment only in using them to the best of our abilities. To settle for less—to live, like less rationally gifted creatures, only at the level of appetite and physical pleasure—is to stop short of claiming our full inheritance as Homo sapiens ("wise man"), which is why our species' stubbornly brutish penchant for clawing after happiness in the form of wealth, fame, pleasure, or power, while it might make for absorbing television, is notoriously unsatisfying and, ultimately, destructive.
Also note that Aristotle says happiness is an activity. It's not a prize to go in search of, something we can discover and thereafter carry around with us or have stuffed and mounted to display above the mantle. Eudaimonia is, rather, something we do with our lives, a way of living that recognizes and works to fulfill the conditions for human flourishing. In other words, while we may not be able to pursue happiness directly, we can live in such a way that happiness may well find us.