The Russian word for "is" is "есть". It is a form of "быть" (to be). In most cases Russian speakers consider it unnecessary to write or pronounce this word. In writing they generally represent the implied "есть" with a dash: "Мяч--красный."
They do sometimes write and pronounce "есть" for emphasis or clarity. For example, if they were arguing about the color of the ball one of them might insist: "Мяч есть красный." This moves the meaning toward: "The ball really is red."
Present tense forms of "быть" other than "есть" are so seldom needed that they are all but forgotten. A few places where they are still encountered:
In the Bible:
Иисус сказал ему: Я есмь путь и истина и жизнь.
Jesus said to him: I am the way and the truth and the life. (John 14:6)
In discussions of philisophy such as in this translation of a lecture into Russian:
Но сегодня говорят «мы». Нынче эпоха самоутверждения масс, а не личностей. Мы есмы. Какое бытие называем мы в этой фразе? Мы говорим также: окна суть, камни суть. Заключает ли это высказывание («мы есмы») в себе утверждение наличности множества «я»? А как обстоит дело с «я был» и «мы были», с бытием в прошедшем? Ушло ли оно от нас? Или мы есмы как раз то, чем мы были? И не станем ли мы как раз тем, что мы есмы?
But today we say "we". This is the era for self-afirmation of the mass rather than the individual. We are. To what kind of being does this phrase refer? We say also: The windows are. The stones are. Does the utterance "We are" imply that there is a multiplicity of "I"? And what about "I was" and "We were", what about existence in the past. Has it left us? Or is it that we are that which we were? And is it not that we will become that which we are? (Introduction to Metaphysics, Martin Heiydegger, 1935)
When a poetic or archaic effect is desired:
О, брат, где ты еси?
O Brother, Where Art Thou? (film title)
The word "is" is a favorite of English speakers. They have found ways to express all sorts of ideas by using "is" to attach labels to things are by declaring that they exist. For example:
The meeting is in that building over there.
To a Russian speaker this sounds odd. He might say:
Собрание проводится в этом здании вон том.
Notice that rather than declaring that the meeting is he has chosen a 'real' verb. He says that the meeting "is being conducted".
Another 'real' verb you can use is "представлять":
Терроризм представляет угрозу мировому сообществу.
Terrorism poses a thread to world society.
You asked about "является". This is another 'real' verb which can be used instead of "есть" in certain cases. The word that comes after it is always in the instrumental case. The word "является" has no exact English translation. It literally means "to appear on the scene as" something. It conveys the idea that under the circumstances which exists, one person or thing acts in a particular role.
We can change change the last example to read:
Терроризм явлается угрозой мировому сообществу.
It is hard to translate this literally in a way that makes sense. This may be close:
Terrorism is on the scene as a threat to world society.
Though in English "is a threat", "poses a threat", or "constitutes" a threat, are good ways to express this idea, they are different expresions. None of them is a translation of "является".
For example:
Кто является директором этого завода?
Who is the director of this plant?
Он сообщил что сосед является террористом.
He informed that his neighor is a terrorist.
Russian speakers might use "является" instead of "есть" in these sentences because "is" here conveys something more than existence. A suit may simple be navy blue, but being a director or a terrorist involves more. They have to be there and do what directors and terrorists do.