What are Sartre's definitions for these three terms "anguish," "forlornness," and "despair"? How does he distinguish among them?
Sartre generally defines anguish, forlornness, and despair as modes of being that extend farther than the mere emotions they represent in humans. He defines them more specifically as follows:
Anguish—Anguish is the awareness of one's own freedom of will and choice. While freedom and anguish are typically not synonymous with one another, Sartre explains the issues that accompany the burden of freedom. There is very little in life that helps us know we are making the right choice when confronted with absolute freedom, and unlimited choice can actually be crippling. Humans desire structure and guidelines, so when we are left to our own devices to make a decision with no framework, the result is a state of anguish.
Forlornness—Forlornness is the realization that God does not exist and that it matters that God does not exist. Such a realization implies that there is no such thing as theoretical goodness and that pure goodness cannot exist either. Additionally, the non-existence of God implies that there is no one responsible for our existence, and thus all of our thoughts and actions have no justifications outside of our own free will.
Despair—Despair is the understanding that one cannot control someone else's actions. Other people are free-thinking and independent creatures, so there is no way to interact with someone else in a way that ensures any sort of certainty in our motives or actions. People other than ourselves will always be unpredictable and unknowable to us. Thus, we can only focus on our own concerns and possibilities because to do so for others is a waste of both time and energy that brings about despair.
The terms identified helps to articulate how Sartre views human consciousness. The sense of "forlornness" results from the rejection of all transcendental and totalizing ideas. When Sartre argues that human beings are "forlorn," he is stressing the idea that they are without any sort of guidance from transcendent ends. They are trapped with only their sense of freedom present, which brings about a natural condition of "anguish," in that nothing nor anyone can help to alleviate the pain of isolation and loneliness, the condition of individual freedom, and more importantly, having to choose. It is this choice that causes "despair" for nothing can lighten the burden of the agony of choice. Take, for example, the student to whom Sartre alludes. Loyal to his country, the student enlists in the French Resistance to the Nazis, but ever loyal to his mother, he realizes that if he leaves to fight for nation, his mother will die alone of a broken heart. The student is poised between equally desirable, but ultimately incompatible courses of action, and represents the essence of human consciousness. He is forlorn, for both loyalty to nation and to mother helps to cancel one another out. One cannot overwhelm the other, making totality a moot point. Additionally, the student is forlorn for nothing can decide for him, only he can, and because of this, anguish and agony results. Freedom becomes brutal, as human beings become actors who are thrust onto a stage to perform without scripts or the aid of a director. Only the glare of a spotlight and the obscure faces in the dark remain.
20th century French philosopher Jean-Paul Sartre, in his seminal essay "Existentialism is a Humanism," (1946) explains that anguish, forlornness, and despair result from existentialism's shift in the responsibility of each human being's existence to the individual himself.
Anguish - According to Sartre, when someone makes a decision, what they are really doing is claiming that the decision they make is the decision every human being should make given the associated circumstances. Sartre refers to the anxiety this induces as anguish, for one's decisions thereby connote immense moral consequences.
Forlornness - Sartre asserts that moral relativity, or the notion that no objective moral standards exist, evokes a certain sense of forlornness. Because of this, humans must decide moral values for themselves. Sartre also refers to this concept as "abandonment."
Despair - Despair, Sartre says, results from the existentialist claim that the existence of God does not influence the existentialist doctrine that human beings must take responsibility for their lives and decide moral values themselves. Sartre views this despair as ultimately optimistic; it catalyzes human beings to begin defining what is moral and meaningful to them and thereby to all human beings.
I hope this helps, and please check out the eNotes guide to Sartre for more information!