Gayatri Mantra

Om Bhur Bhuvat Svah
Tat Savithur Vareniyam
Bhargo Devasya Dhimahi
Dhyo Yo Nah Parcho Dayat

Which means: Let us meditate on the splendor of the God Savitru (Sun), who will illuminate our understanding.

Ashtanga Invocation Chant

The chant done in call and response, slowly so it is easy to learn:

OM
VANDE GURUNAM CARANARAVINDE
SANDARSITA SVATMA SUKHAVA BODHE
NIH SREYASE JANGALIKAYAMANE
SAMSARA HALAHALA MOHASANTYAI
ABAHU PURUSAKARAM
SANKHACAKRASI DHARINAM
SAHASRA SIRASAM SVETAM
PRANAMAMI PATANJALIM
OM

Which is roughly translated into English as:
I bow to the lotus feet of the gurus, The awakening happiness of ones own self revealed, Beyond better, acting like the jungle physician, Pacifying delusion, the poison of samsara. Taking the form of a man to the shoulders, Holding a conch, a discus, and a sword, One thousand heads white, To Patanjali, I salute.

Ashtanga Closing Chant

OM
LOKAA SAMASTHA SUKHINO BHAVANTHU
OM SHANTI SHANTI SHANTI
OM

Lokaa: The location of all universes existing at this moment
Samastha: All beings living in this location
Sukhino: In happiness, joy and free from all suffering
Bhav: The divine mood or state of union
Anthu: May it be so

Prayer for Enlightenment

OM
ASATOMA SADGAMAYA
TAMASOMA JYOTHIRGAMAYA
MRITHYORMA AMRUTANGAMAYA
OM SHANTI,, SHANTI, SHANTI

Lead us from darkness to light
From ignorance to truth
And from death to eternity
Let peace prevail everywhere