You may have heard a yoga teacher say that Ujjayi means "victorious." The word "ujjayi" is derived from the Sanskrit root "ji" (जि) with the prefix "ud" (उद्) added to it. So the combined root is "ujji" (उज्जि) which means "to be victorious".
Ujjayi (pronounced "oo-jai") is called "the victorious breath" because the dedicated practice of this pranayama will create victory over disturbances in the breath, which will lead to victory over disturbances of the mind.
The practice of Ujjayi will clearly highlight tensions in the breath, in the muscles of respiration, in the area of throat and chest, and in the mental anxieties that arise upon watching the breath and finding these disturbances. By steadily bringing your attention to these disturbances and working on lengthening and smoothing out the breath, the mind relaxes.
Once a practitioner is more versed in the practice of Ujjayi and the physical disturbances are not as pronounced, the practice becomes a touchstone that helps you sense where in the body-mind you are holding tension or not fully present.
Ujjayi is an anchor for the attention. By anchoring the attention to the breath, the mind becomes softer and more relaxed. As the breath becomes smoother and freer, so does the mind. And this is why Ujjayi breath is called "the victorious breath."
You can learn Ujjayi breath from my Forrest Yoga video tutorial for Yoga International or via my free Soundcloud audio tutorial embedded below. Happy breathing!
P.S. Luke Ketterhagen also offers a great learning video on Ujjayi. However, I disagree with him a bit on the topic of when not to use Ujjayi. In Forrest Yoga, we do practice Ujjayi through the majority of each practice. The breath is woven throughout to help each practitioner stay focused and in a feeling state.
P.P.S Check out my "Experience Forrest Yoga" video series at Yoga International. Four classes that starts with introduction to the basic moves of Forrest Yoga, including Ujjayi pranayama.