To my limited knowledge, in Zen, it is believed that anyone can be enlightened(if dedicated enough). Does everyone have this potential in other branches of Buddhism? Can everyone achieve this through meditation? Is this what is meant by Buddha nature? Is it exclusive to humans(is language necessary?)
Buddhist philosophy, or even philosophy in general makes words difficult to use consistently. The concept of everyone is one of those words. Which makes the answer yes and no to the question: Can everyone be enlightened?
In, I believe, all schools and branches of Buddhism, all sentient beings can eventually gain enlightenment. With emphasis on "eventually".
Animals are not able to, as long as they are an animal.
Dwellers of lower realms, have an extremely hard time becoming enlightened so, and would probably need to accumulate some positive karma to become a human before they have a chance to.
Beings of higher realms like deva's tend to have such a happy and fun life, practising is difficult, but not impossible.
The human realm is seen as the best of all worlds in terms of practice.
Humans that have committed "Anantarika-karma" or Heinous Crime, (matricide, patricide, killing ahrat, wounding Buddha, creating a schism in the sangha) can no longer achieve enlightenment in this life.
Can everyone achieve this through meditation?
Now this is a difficult one. In the texts some of the enlightened students became so by listening to the words of the Buddha. Others became through practise. The main attributes needed seem to be a pure mind and insight, whether gained by meditation or otherwise.
Is this what is meant by Buddha nature?
I think it is, to an extend. This seems to be mainly a Mahayana concept and I'm not that familiar with it. I believe it's considered literally: "the potency to become a Buddha" In the books I read of the Dalai Lama it is mainly revered to with respect to developing loving-kindness: "Training or developing your Buddha nature."
Is it exclusive to humans(is language necessary?)
Heavenly beings can also practise and become enlightened. I'm uncertain whether language is necessary, I can remember no sutta's about the subject.
It is my understanding from reading Dogen that everyone already has Nirvana within them and it is hidden by a mask and revealed through sincere practice http://www.tricycle.com/web-exclusive/fundamentals-dogens-thoughts?page=0,1
Dogen says, “Between aspiration, practice, enlightenment, and nirvana, there is not a moment’s gap.” Thus, nirvana is one of the four elements in a practitioner’s spiritual activity. For Dogen, nirvana is inseparable from enlightenment, and it is inseparable from one’s practice at each moment. In other words, there is no authentic practice that lacks enlightenment or nirvana.
While Dogen discusses aspiration, practice, and enlightenment in detail, he does not explain the last element, nirvana, which seems to be an invisible element in his teaching. It is as though he talks about the experience of nirvana without using this word.
Nirvana is regarded as the realm of nonduality, where there is no distinction between large and small, long and short, right and wrong, appearing and disappearing, self and other. It may be called reality itself, or the absolute place beyond time and space. This is a realm that cannot be grasped objectively. The intuitive awareness or transcendental wisdom that goes beyond dualistic, analytical thinking and leads us into this realm is called prajna in Sanskrit.
Dogen calls this place of inner freedom the buddha realm. It is where one is many, part is whole, a moment is timeless, and mortality is immortality. To experience this beyondness in the midst of the passage of time, change, and decay is a miracle. For Dogen, this miracle can happen each moment, as each moment of duality is inseparable from a moment of nonduality.
Dogen is generally considered to be a Zen practitioner, but how can we exclude his observations from his own experience from any Buddhist practice?
Here is another source that reinforces the ever present Nirvana in each of us http://bdtest1.squarespace.com/web-archive/2010/9/9/moment-by-moment-nirvana.html
It is not that duality stops existing or functioning; the small is still small and the large is still large. The body remains the body and the mind remains the mind. Without discerning the differences between things, we could not conduct even the simplest task of our daily lives. And yet, in meditation the distinctions seem to dissolve and lose their usual significance. Dogen calls this kind of nondualistic experience nirvana, which exists at each moment of meditation.
There are so many areas of Buddhism and sects and circles within each sect that to find a universal belief in every sect of Buddhism is doubtful. My primary concern is does the primary area of my practice hold this to be true, or if nothing else do I hold this to be true. My own practice confirms to whatever degree possible what Dogen says from his ultimate experience.
Anybody has the ability to become enlightened however different parts of Buddhism have different beliefs on how this happens. Everybody already has the seed of Buddha hood/ Nivada inside them which is hidden in till revealed in sincere practice. As long as the person is committed to becoming enlightened they have the ability to do so. Animals are the only exception to this rule as they need to create good Kama to enter the human realm before haveing the ability of enlightenment. People of higher realmes have to exercise restraint to avoid becoming attached to impermanent objects which cause Dukka (dissatisfaction) to exit the cycle of death and rebirth known as samsara. While this is the case in Zen Buddhism pure land Buddhism believes that you must have a relationship with Amitabha Buddha whereby one must call their name or chant homage. Pure land believers then hope to be reborn in samsara in the pure land were it is easier to stain enlightenment.
This is my understanding which is have understood from my GCSE course however it may not all be correct.
Self effort and meditation to achieve enlightenment are not universal in Buddhism. In Pure Land schools of Buddhism one has a relationship with Amida/Amitabha Buddha whereby one calls their name or chants homage to their name. Such a believer would aspire to be reborn in the Pure Land where one can much more easily become enlightened as one would then be hearing the Dharma directly from Amitabha Buddha. More information on Amitabha.