The Englishman is important to Paolo Coelho's The Alchemist because he acts as a foil to Santiago. Both men are on a quest, but the Englishman is going about it the wrong way.
A foil is another character to which the protagonist is compared. They are similar in some ways, but very different in areas that are central to the story's plot. A foil is...
...[a] character who sets off another character by contrast.
As an example, Hamlet's foil in Shakespeare's play by the same name is Young Fortinbras. Both have lost fathers, both uncles have taken over the throne, and both want revenge; but where Fortinbras is chided by his uncle to behave himself and move on in order to take some valuable action (like fighting another country), Hamlet cannot decide either way, and he beats himself up over it. (And whereas Fortinbras' father died in a fair fight with Old Hamlet, Old Hamlet's brother murdered him.) The comparison of the two makes Hamlet's character stand out more clearly.
In The Alchemist, Santiago (the boy) is searching for something: he wants to discover his Personal Legend—what he was meant to do and the person he was meant to be. The Englishman is also searching. He wants to learn how to be a great alchemist, and believes that by visiting the alchemist in the desert, he will find a way to turn common metals into gold.
The Englishman is very different because he believes he can find the answers in books. As he travels the desert with Santiago, he carries a number of books with him, even though they are traveling by camel. He stresses to Santiago the importance of reading to find his answers. Santiago, on the other hand, tells the Englishman to look for signs—omens—in the desert as they travel. Each agrees to practice the other man's theory on the trip. Ultimately, the books do not help Santiago; and, the Englishman finds nothing of value in the desert. This is where we see the enormous difference between the two men. Santiago learns to speak the Language of the World (watching for omens) and to "trust the Soul of the World." In this way, he is eventually able to become one with the universe. He is able to "command the wind" or so the chief of the desert tribe believes! The alchemist tells Santiago:
Don't give into your fears...If you do, you won't be able to talk to your heart.
Being able to tap into his heart rather than his mind is essential for Santiago's success. The Englishman does not rely on his heart—simply on learning and information. This is not the path of one who wants to live his Personal Legend. Because the Englishman has not discovered this, he will continue to search without success. The alchemist gives Santiago perhaps the most important advice there is:
If a person is living out his Personal Legend, he knows everything he needs to know. There is only one thing that makes a dream impossible to achieve: the fear of failure.
The Englishman is important to the story in that the reader can see the true path and the one that does not work. Relying on one's brain does not bring happiness. Relying on one's heart is of the utmost importance. It is by comparing the two characters that Coelho shows the reader how he or she should live his or her life. This story is allegorical. In other words, Coelho is attempting to teach a lesson—to tell the world that knowledge does not lead to joy or success, but listening to one's heart and being one with the world will bring everything necessary for happiness.
The Englishman is an important character because he is also on a personal quest, and he provides a positive role model.
The king tells Santiago that everyone needs to find his own personal question. When Santiago dreams of a child who takes him on a magical journey to the Egyptian pyramids, he decides to sell his sheep and go off on his quest. The desert is a difficult place, but he runs into the Englishman. The Englishman is also on a personal quest.
All his life and all his studies were aimed at finding the one true language of the universe. (p. 68)
The Englishman does introduce Santiago to alchemy, but he is more important as a spiritual guide. The Englishman is educated and passionate. He is also a real, physical person and not a vision. He reinforces what the soothsayer and the vision of the king tell Santiago: follow your dreams.
"The Englishman serves as a contrast to Santiago’s way of learning. Santiago meets him when the two are preparing to join the caravan. The Englishman is desperately seeking the famous alchemist who is purported to live at the Al-Fayoum Oasis. He thinks that all of his knowledge will be found in books and does not learn from experience the way that Santiago does. When Santiago returns the books he has lent, he reacts in a bitter way, thinking to himself that 'his soul must be too primitive to understand those things' found in the learned tomes."
He is important because you are able to see the different ways they see and go about their quest. The Englishman listens mostly to books while Santiago listens to his heart. Coelho is trying to tell us that listening to our heart will bring happiness.