Zhab-tens are prayers for the long life of our Teachers. Literally, the word zhab-ten means "feet stable," as in "May the feet of [this or that] Teacher be secure or stable." Whenever we do sadhana, the main body of the practice may be described by the three excellences, the dampa sum (also known as the jor-nu je-sum). These are 1) giving rise to Bodhisatvic motivation, 2) carrying out the practice as well as possible, and 3) dedicating the merit from that practice for the sake of all sentient beings. We are taught that, within tantric Buddhism, these are the three things that make a practice perfect or excellent. However, in the actual day-to-day practice of Vajrayana Buddhism, the three excellences are routinely followed by two other elements, zhab-ten (Long-life prayers) and mon-lam (aspiration prayers).
Actually, zhab-ten are a special kind of aspiration prayer. We have just accumulated vast stores of merit through our tantric skillful means. Now we want to ear-mark exactly how that merit is going to come to fruition. So we say aspiration prayers. "May this or that happen." "May this or that come to pass." "May there be... [Fill in the blank]." However, within Vajrayana, the Teacher is the root of our practice. Without the Teacher, there simply is no Vajrayana practice. Therefore, because no matter what we do, in Vajrayana we should always remember the Guru first, the first kind of aspiration prayers we say are zhab-ten.
Most Tibetan Buddhist Teachers have zhab-ten, and certainly all Tibetan Teachers Who are tulkus. The way zhab-ten are usually written is that a student of a particular Teacher goes to that Teacher's Teacher or high lineage-holder and requests that other Teacher to write a zhab-ten for his or her Teacher. this is usually accompanied by a khata and, these days, a monetary offering, sometimes quite considerable. It is also possible for a well-known Guru to write a zhab-ten for one of His or Her students Who is also a Teacher without having to be asked as a spontaneous gesture of appreciation and respect for that student. In either case, the zhab-ten is usually composed according to traditional poetic conventions, for instance including the Teacher's name within the metaphors of the verse. So if a Teacher's name is Pema (Padma, lotus), then there will be at least one lotus mentioned in the zhab-ten. Or if the Teacher's name is Dorje (vajra), there will be at least one thing that is vajra-like mentioned, and so on till the Teacher's full name has been incorporated into the prayer. The zhab-ten will also end by requesting that the Teacher's feet remain secure for hundreds of eons, until samsara is ended, or something along those lines.
When Lord Buddha left for Tushita Heaven in order to teach the Dharma to His mother, Mahamaya, His students back on earth offered up unrelenting and heart-wrenching prayers for the Tathagata to return, and return He did after three months in answer to their prayers. Similarly, when we say the seven (or eight or 10) limb prayer (yen-lag dun-pa) for accumulating merit and purifying sin, requesting the Teacher to remain in this body and realm is one of those seven limbs. Teachers, and especially tulkus, are here specifically for our benefit. If no one wants Their services, then there's no reason for Them to be here, and so They may choose to go to some other realm where They are wanted and needed. In Vajrayana, it is important for students not to think of our Teachers as ordinary human beings. Our samaya includes our pure perception (dag nang) of our Teachers, seeing Them as Guru Vajradhara, actual Buddhas here and now, and high-level Bodhisatvas Who are not subject to birth and death the same way we are.
In Vajrayana, we are not actually supposed to pray for our Teachers. As Buddhas, They don't need our help. Rather, it is we who need Theirs. So zhab-tens are not prayers for our Teqchers' long life. They are prayers to our Teachers to remain for a long time for our sake. This is an important distinction. We are supplicating the Teacher to remain in His or Her body. We need Them to be here so They can continue to teach us, discipline us, bless us, and empower us. In my experience, the more time we can spend with our Teachers, as closely as possible and with proper samaya, the better will be our practice and the quicker its result. These days, many practitioners have scant face-to-face contact with Their Teachers, and, in my opinion, this is one of the reasons we don't see a lot of highly developed modern practitioners. Be that as it may, if our Teachers leave Their human bodies, then we really have no chance to benefit from Their Nirmanakaya, or at least not until They take rebirth, are discovered, and trained, a process that may only be complete when we ourselves are about to die.
I had the good fortune (and sometimes the extreme discomfort) of living in close contact with my Teachers for the better part of 20 years. Right from the very beginning, we never completed any practice, from offering a cup of water or lighting a candle, to offering a seven-day, full-on drubpa, without immediately saying the zhab-tens to my Teachers three Root Gurus and another general zhab-ten by Khyentse Cho-kyi Lodro to cover all Their other Teachers. We then followed these zhab-ten with a half-dozen or more mon-lam or aspiration prayers. Looking back on this discipleship, I can't tell you how grateful I am that I was trained this way. So often today, we Western practitioners conclude our practices with a scant mon-lam or two, with zhab-tens being reserved for special teachings and ceremonies, if they are said at all. Personally, I find this less than "best practice."
As stated above, in Vajrayana, it is important to constantly remember the Teacher -- 24/7/365 if possible. The Teacher embodies the Buddha, Dharma, and Sangha. He or She is the source of our instruction and the sine qua non of tantric practice. We are supposed to think first of our Teacher(s) in every situation, from passing out tshog or even prayers, books, and coasters, to entering a door, getting into a car, or stepping out ahead on a walk. Routinely saying one's Teacher's zhab-ten is not only a way to insure our Guru's continued bodily presence, but also helps us develop that constant remembrance of and praying to our Teachers. It is a great habit to cultivate and I recommend it highly. If our Teachers are Tibetans, it is a courtesy to Them (and another way of thinking of Them first), to say Their zhab-ten in Tibetan. However, at home, we can say our Teacher's zhab-ten in either or both languages. The important thing is to make saying our Teachers' zhab-tens as routine a tying our shoelaces.
In terms of the benefits of saying our Teachers' zhab-tens, of course there is our Teachers' remaining in this world for the sake of the Teachings and all sentient beings. However, based on interdependent connection, saying our Teachers' zhab-tens also results in accumulating our own long-life merit. So not only does saying our Teachers' zhab-tens help insure our Teachers' long life, it also helps insure our own. Bottom line, if you're not currently saying your Teacher's zhab-ten after every act of Dharma, I highly encourage you to begin.