TRYAMBAKKAM refers to the Three eyes of Lord Shiva. 'Trya' means 'Three' and 'Ambakam' means eyes. These three eyes or sources of enlightenment are the Trimurti or three primary deities, namely Brahma, Vishnu and Shiva and the three 'AMBA' (also meaning Mother or Shakti' are Saraswati, Lakshmi and Gouri. Thus in this word, we are referring to God as Omniscient (Brahma), Omnipresent (Vishnu) and Omnipotent (Shiva). This is the wisdom of Brihaspati and is referred to as Sri Duttatreya having three heads of Brahma, Vishnu and Shiva.
YAJAMAHE means, "We sing Thy praise".
SUGANDHIM refers to His fragrance (of knowledge, presence and strength i.e. three aspects) as being the best and always spreading around. Fragrance refers to the joy that we get on knowing, seeing or feeling His virtuous deeds.
PUSTIVARDHANAM Pooshan refers to Him as the sustainer of this world and in this manner, He is the Father (Pater) of all. Pooshan is also the inner impeller of all knowledge and is thus Savitur or the Sun and also symbolizes Brahma the Omniscient Creator. In this manner He is also the Father (Genitor) of all.
URVAAROKAMEVA 'URVA' means "VISHAL" or big and powerful or deadly. 'AAROOKAM' means 'Disease'. Thus URVAROOKA means deadly and overpowering diseases. (The CUCUMBER interpretation given in various places is also correct for the word URVAROOKAM). The diseases are also of three kinds caused by the influence (in the negative) of the three Guna's and are ignorance (Avidya etc), falsehood (Asat etc as even though Vishnu is everywhere, we fail to perceive Him and are guided by our sight and other senses) and weaknesses (Shadripu etc. a constraint of this physical body and Shiva is all powerful).
BANDANAAN means bound down. Thus read with URVAROOKAMEVA, it means 'I am bound down by deadly and overpowering diseases'.
MRITYORMOOKSHEYA means to deliver us from death (both premature death in this Physical world and from the neverending cycle of deaths due to re-birth) for the sake of Mokshya (Nirvana or final emancipation from re-birth).
MAAMRITAAT means 'please give me some Amritam (life rejuvinating nectar). Read with the previous word, it means that we are praying for some 'Amrit' to get out of the death inflicting diseases as well as the cycle of re-birth.
Now you are ready to recite the Maha Mrityunjaya Mantra