How do Plato and Aristotle define the good?
Plato defines "the good" as an unchanging "form" that cannot be comprehended by sight or other senses. There were other forms, like "truth" and "beauty," but the "good" was the highest of these forms. Certain things, people, concepts, and so on could have "goodness," but they could not be good in and of themselves, because people might disagree about their goodness, or, more fundamentally, they could be destroyed, by death or forgetting. So the only thing that was permanent was the form or the "idea" of goodness that could only really be comprehended by philosophers who could transcend the fleeting knowledge provided by the senses.
Aristotle understood the good in less abstract ways. He thought that the "good" was contingent on situations and the individual. What was important to him was that a person's ethical understanding informed their actions. Rather than an absolute, abstract concept, Aristotle understood "the good" to be a "good life," and actions were "good" to the extent that they furthered this end. As the philosopher himself writes in Book One of the Nicomachean Ethics, it is not terribly meaningful to assign "goodness" as a form or idea:
Let us again return to the good we are seeking, and ask what it can be. It seems different in different actions and arts; it is different in medicine, in strategy, and in the other arts likewise. What then is the good of each? Surely that for whose sake everything else is done. In medicine this is health, in strategy victory, in architecture a house, in any other sphere something else, and in every action and pursuit the end; for it is for the sake of this that all men do whatever else they do.
In a sense, one could answer this by saying that while Plato struggled to define "the good", Aristotle resolved the problem by saying that "the good" in and of itself does not exist.
The essential distinction has to do with Plato being concerned about the Forms, the notion of a deeper reality of abstract ideals of which the phenomena we perceive with our senses are merely faint shadows or imitations. For Plato, all things we consider good participate in the Form of the Good.
For Aristotle, "separated" forms did not exist. Goodness was simply the commonality we perceive among things we consider good and of which we approve. In his ethical works, he suggests that we are naturally drawn to good things and thus we use the term goodness to describe them. For Aristotle, the highest human goods are those we seek for themselves alone, rather than as means to something else. Thus, he would say that the highest human good or type of well-being ("eudaimonia") is the virtuous life.
In technical terms, Plato is a "realist", one who considers that abstractions possess some type of independent reality, while Aristotle is a "nominalist" who argues that our abstract terms are merely shorthands for perceived commonalities and have no independent existence.
Both philosophers make the argument that "the good" is something that must be pursued as part of one's nature as a human being. At the same time, both of them understand this pursuit as something that must take commitment and a sense of courage in order to embrace and, eventually accomplish. Yet, both thinkers define "the good" in different ways. For Aristotle, this is an internal quest. Aristotle believes that every human being has a "good" end within them and they must recognizes this. Ethical conduct comes out of this understanding. It is here where Aristotle defines this pursuit as the "golden mean" where individuals recognize good as the balance between two ends. For example, ambition is good, but too much can be bad. Ethical conduct for "the good life" is found by the individual in this moderation. Plato differs in his notion of "the good." Plato defines "the good" as the embrace of "the form," the ultimate good. This is an external reality that Plato readily admits not everyone can achieve. The "form" is an ultimate good that defines "the good." In this, Plato's conception of the good is something that is divergent from Aristotle's.