The meditation that we do in Seido Karate, and that's usually found connected with Japanese martial arts in general, is in the Zen style.
We say "Zen meditation", but technically that's redundant. "Zen" is the Japanese pronunciation of "Ch'an", which is the Chinese pronunciation of "dhyana", which is a Sanskrit word meaning "meditation". So "Zen meditation" literally means "meditation meditation"! But we usually mean it to mean a specific style of meditation: one that developed out of a certain school of Buddhism, but since the 13th century became more widely distributed throughout Japanese culture, and in the 20th century spread throughout the world.
So Zen is not the only type of meditation. For example there's Transcendental Meditation. That was big in the 60s, it was what the Beatles traveled to India to study, and seems to be enjoying a bit of a revival. Advocates claim that it will relax you or even bliss you out. (John Lennon talked about how the people he studied with acted like they were in a competition "to reach God quicker than anybody else", to see "who was going to get cosmic first." Lennon later realized, "What I didn’t know was I was already cosmic.") The Catholic tradition has its own type of meditation, really a form a prayer seeking contact with God -- definitely not what we do in zazen. "Mindfulness meditation" for stress reduction has become popular in recent years, and there are also guided meditations which are supposed to promote health or have various other benefits.
There are forms of mediation that arose out of other forms of Buddhism and are rather different than that found in Zen. There are Hindu meditation practices that are older than Buddhism, the ones that the Buddha practiced and mastered (according to scriptures of the Buddhists, at least) before deciding that they weren't helpful in his quest.
Zen meditation contrasts with most of these other types in two important ways. First, Zen teachers are adamant that zazen is not about reaching some special state of consciousness or about trying to direct the mind to some specific content. You're not trying to bliss out or know the mind of God or reach a state of enlightenment. "Buddha-mind is your everyday mind" is almost a stock phrase in Zen. Dogen, the founder of the Soto style of Zen, said that just sitting, if we do it whole-heartedly, is enlightenment itself. In Zen meditation there is nothing more to seek, nothing to reach for, nowhere to go. You're already there. Like those old Palmolive commercials, "You're soaking in it". Just sit.
Second, Zen tends not to give very detailed concrete instructions about how to meditate. In TM, for example, there's something to do, a mantra you're supposed to chant. In guided meditations there's something you're supposed to visualize. I don't know much about vipassana "insight" meditation from Theravada Buddhism, but from what I've read there are various contemplations that you are lead through as part of the practice. Zen doesn't have much of this: it tells you what posture to sit in, gives you a few generalities, and you're on your own.
This lack of instruction is somewhat related to the first point -- if you're not trying to do something or go somewhere, it's hard to give you detailed instructions! If you go to Google Maps and type in your starting address as your destination, you don't get a very long response. Inherent in the idea that you need a long set of detailed instructions to meditate is the idea that there's somewhere to go, that your meditation mind is not the same as your everyday mind. Zen disagrees. (Some days I think this disagreement is more semantic than actual, and some days it seems tremendously important. I don't know.)
But this lack of instruction can be tricky, even frustrating. A Soto Zen nun named Gesshin recently wrote on her blog:
I've tried to coerce teachers into giving me more instruction than that, but they never do. I was really frustrated with this for years. What the hell am I supposed to be "doing with my mind?" Actually, I still think that sometimes.
Zen teacher Brad Warner concurs:
But as a long-time zen practitioner the idea of "meditating correctly" strikes me as deeply absurd. Yet, also as a person who has undergone "years (of) being uncertain whether he is meditating correctly" I understand the appeal of something that’s supposedly going to give you solid answers rather than being merely "vague and paradoxical."
Here we have two people whom we would expect to be knowledgeable about Zen, talking about being uncertain if they're doing it right. What the heck? That might seem strange if we think about mediation as a skill to learn, the same way we think about knowing how to change your oil or how to swing a golf club.
But this "not knowing" is a form of shoshin, "beginner's mind", a very important idea in Zen and in the martial arts. The Zen master Shunryu Suzuki famously said, "In the beginner's mind there are many possibilities, in the expert's mind there are few." "Beginner's mind" is another way of saying "I don't really know what I'm doing." And that's good! It's wonderful! If you don't "know", then you can keep looking, you can keep learning.
Another way to look at these two ideas of not reaching for a special state and not giving detailed instructions, is the idea of witnessing. Perhaps we could say that doing zazen is bearing witness to your mind.
And a good witness must be unbiased. If you ever watch courtroom dramas, sometimes you'll see lawyers object that someone is "leading the witness" -- that is, they are asking very specific questions in order to get the witness to answer a certain way and draw a conclusion. But that's not very good witnessing. It makes the witness, unconsciously, tend to bias and filter their testimony.
In the same way, if you have a long list of instructions for your meditation practice, or tell yourself that you should go into some sort of special state, that will determine what you see as you sit. You're leading the witness, and so won't accurately see your own mind.
But it's tricky, because to even label the thing as "witnessing" is a bias, leading yourself on. So don't take that too seriously.