Tapas is one of yoga's Niyamas. The Niyamas form the second of Yoga's Eight LImbs, and include five principles that come together to form the "observances" of a yogic life style.
Tapas is a pretty hardcore Niyama. It is understood as fire, catharsis, self-discipline, or austerity.
In her lovely book on the Yamas and Niyamas, Deborah Adele explains why one would want to endure or pursue Tapas, "Tapas eventually changes our nature, turning us into a cauldron that can withstand any of life's challenges" (134).
Using the fire metaphor associated with Tapas, I like to think of Tapas as akin to the process of tempering metal. By mindfully applying a little heat, we can strengthen our constitutions.
As I said, this is where the practice gets hard core. Though one needn't practice sleeping on a bed of nails to invoke tapas, even the simple enforcing of discipline on a day can bring discomfort and requires developing strength. For example, adhering to daily meditation, journaling, and neti pot use could become a disciplined ritual.
In a yoga pose, this can mean simply staying in the discomfort, as in chair pose, and, as one of my teachers (the wonderful Zachary Cannady) instructs, simply watching the fire move through the body. Feeling the legs and arms shake, you stay there, become stronger, becoming more able to live calmly through instability and insecurity.
The idea here is that by consciously and safely practicing Tapas, we steel ourselves (to stay with the tempering metaphor) for life's ups and downs, and become stronger in the process.
For example, I have a colleague who gives her graduate students a midterm, asking them to sit and write for four hours straight. They hate the experience, but they appreciate that this painful, firey time helps prepare them for the much more extreme experience of comprehensive exams.
In my own writing practice, I have gotten pretty comfortable with writing for 25-30 minute bursts. Now I am trying to gradually extend how long I sit and to observe what makes me get up. I am learning that I am most undisciplined when revising drafts, particularly when it comes to writing new sentences at that stage, and particularly when those new sentences are revisions. As I sat working on an article yesterday, I discovered myself looking at the clock in those moments. Through the practice of Tapas, I learn where I am weak, and try to stay in it, feeling the burn.