I am confused about the process of casting a vote for someone who helped me (he is a good teacher). How can this be done?
As a new user, the best way to reward the person whose answer helps you the most with your question is to accept their answer on your question. How to do this is explained in the FAQ under "How do I ask questions here?". This lets other people know you have received a good answer, and give a reputation reward (15 rep) to the person who posted that answer. Accepting your answers also gives you a small reputation reward (2 rep).
Once you have at least 15 reputation, you can also vote up questions and answers that you think are especially useful. This also gives the poster a reputation reward for providing a useful post (up-votes give 5 rep to someone asking a useful question, and 10 rep to someone providing a useful answer).