The Russian word for "is" is "есть". It is a form of "быть" (to be). In most cases Russians 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. For example if they were arguing about the color of the ball one of them might insist: "Мяч есть красный." "The ball really is red."

In certain cases "is" is translated "является". However "является" does not literally mean "is". Instead it means "appears" or "shows up". It is used to say that someone or some thing is acting in a particular role or capacity. The role or capacity is always in the instrumental case in order to convey the idea "as a" or "as the".

For example:

Он является директором. (He is the director.)

Он является террористом. (He is a terrorist.)

It is appropriate to use "является" here because these are roles which these men have assumed and which they can put off. The director was not born a director and may die a former director. He is acting in his role as director.

In contrast the sentence "Мяч является красным." makes no sense. It suggests that the ball's words are deeds are what make it red. Say "Мяч--красный." If someone contradicts you, say "Мяч есть красный."