TARE means liberating from all discontent.
TUTTARE means liberating from the eight fears, the external dangers, but mainly from the internal dangers, the delusions.
TURE means liberating from duality; it shows the true cessation of confusion.
SOHA means "may the meaning of the mantra take root in my mind."
In Tibetan, Tara is known as "Dölma", or "She Who Saves." In particular she represents compassion in action, since she's in the process of stepping from her lotus throne in order to help sentient beings.
Tara's mantra is a loving play on her name. According to Sangharakshita, a traditional explanation of the mantra is that the variations of her name represent three progressive stages of salvation:
1. Tāre represents salvation from mundane dangers and suffering. Tara is seem as a savioress who can give aid from material threats such as floods, crime, wild animals, and traffic accidents.
2. Tuttāre represents deliverance into the spiritual path conceived in terms of individual salvation. In traditional terms, this is the path of the Arhant, which leads to individual liberation from suffering. This is seen in Mahayana Buddhism as a kind of enlightenment in which compassion does not figure strongly.
3. Lastly, ture represents the culmination of the spiritual path in terms of deliverance into the altruistic path of universal salvation - the Bodhisattva path. In the Bodhisattva path we aspire for personal enlightenment, but we also connect compassionately with the sufferings of others, and strive to liberate them at the same time as we seek enlightenment ourselves.
Svaha, according to Monier Monier-William's Sanskrit Dictionary, means: "Hail!", "Hail to!" or "May a blessing rest on!" Her mantra can therefore be rendered as something like "OM! Hail to Tara (in her three roles as a savioress)!" although this may one of those occasions when the mantra is best left untranslated.