"Hai!" means "Yes!" or "I'll try!" when used in response to an instructor's advice. It is more energetic and polite than the English expressions "Okay," or "Yeah."
"Rei" means etiquette or bow. Although you usually only hear the term when asked to bow during the opening and closing ceremonies at each class, the entire experience of training at a traditional dojo should be one of courtesy and joy.
"Onegai shimasu" is used when we bow to one another, and is a request to cooperate in training. In some schools, the expression is shortened to "Osu!" and can be used to express one's enthusiasm for training.
"Arigato gozaimasu" is a polite way of saying thank you. We generally say it to the instructor at the end of the last bow when closing class. "Arigato" by itself is too casual for this setting. "Domo arigato gozaimasu" means thank you very much, and is also acceptable to use when addressing the instructor.
"Sensei" means teacher. Append it to the end of an instructor's last name, as in "Smith-Sensei." The suffix "-san" is appropriate between peers ("Smith-San"), but should not be used when addressing a teacher.
More explanations of these words can be found in the December 8, 2006, post entitled "Your First Day in the Dojo."
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