What is the meaning of "Namaste"?

I say "Namaste" after the end of each yoga class, with my palms together in front of the heart, but I often forget to tell a group of beginners what this means.

Simply, "namaste" means "I bow to the divine in you".   It's a friendly greeting of respect commonly used in India that has been adopted by many Westerners who practice yoga or other spiritual and health practices that derive from the Hindu culture.

In Sanskrit, "Namas" means, "bow, obeisance, reverential salutation."  "Te" means "you".   So, "I reverentially bow to you."

The hand position, anjali mudra,  of pressing the palms together with fingers pointing upward, has the same meaning as "Namaste" and can be done silently without the word itself.

A few of the more poetic translations are (from Wikipedia "Namaste"):

• "I honor the Spirit in you which is also in me."
• "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."
• "I salute the God within you."
• "Your spirit and my spirit are ONE." 
• "That which is of God in me greets that which is of God in you."
• "The Divinity within me perceives and adores the Divinity within you."
• "All that is best and highest in me greets/salutes all that is best and highest in you."
• "I greet the God within."

The following is from Aadil Palkhivala, in a Q&A segment in YogaJournal.com on the Meaning of Namaste

"Ideally, Namaste should be done both at the beginning and at the end of class. Usually, it is done at the end of class because the mind is less active and the energy in the room is more peaceful. The teacher initiates Namaste as a symbol of gratitude and respect toward her students and her own teachers and in return invites the students to connect with their lineage, thereby allowing the truth to flow—the truth that we are all one when we live from the heart."

(Aadil Palkhivala is a nationally known yoga teacher who has strong underpinnings in the Iyengar Yoga tradition.   He is also a naturopath and an ayurvedic practitioner.)