The spoken "Namaste" (and the gesture, Anjali Mudra) represent the belief that there is a Divine spark within each of us or also can be interpreted as honoring the shared spirit or oneness of humanity. I have always liked the translation, "I honor the place in you in which the entire universe dwells, I honor the place in you which is of love, of integrity, of wisdom and of peace." When you are in that place in you, and I am in that place in me, we are One."
"Nama" means bow, "as" means I, and "te" means you. Therefore, Namaste (formed through external sandhi) literally means "bow me you" or "I bow to you."
To perform Namaste, place the hands together at the heart chakra (heart center), close the eyes, and bow the head. It can also be done by placing the hands together in front of the third eye, bowing the head, and then bringing the hands down to the heart (considered an especially deep form of respect). Pretty nifty, right?
The principle behind the movement is that we bring the hands together at the heart, acknowledging love, in conjunction with bowing and closing the eyes acknowledging surrender and respect.
Within a class, Namaste allows teacher and student (as well as the class as a whole) to show gratitude and reverence to one another. The teacher initiates Namaste as a symbol of honor toward his/her students and his/her own teachers and in return invites the students to connect to the idea that we are all one when we live from the heart.
Even as somewhat of cynic by nature, I appreciate the beautiful sentiment in Namaste. We are all a part of some shared experience, linked. Acknowledging our connectedness and appreciating someone else's journey is so simple, but has the potential to be incredibly powerful and all it takes is a bow.
नमस्ते (Namaste in Sanskrit)