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In mathematical philosophy, one asks the question "do mathematical objects really exist"?

This is then followed by "yes" or "no" answers, but does the question even make sense? Is it even meaningful to talk about the existence of an idea? Of a concept? Of a equation?

So basically, that's my question. When philosophers talk about whether mathematics is real or not, what definition of 'real' are they using? What definition of 'exist' do they use to judge whether mathematical objects exist or not?

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You are, in a sense, begging the question against (a part of) those who ask themselves the question whether mathematical objects really exist. That is, because you already come equipped with a certain theory of what mathematical objects are. According to you, mathematical objects are ideas, concepts and equations.

A big part of the discussion about the existence of mathematical objects concerns the question what mathematical objects could be. Can we have mathematics as we practice it today if mathematical objects are concrete? Can we have knowledge of mathematics when mathematical objects are abstract? If so, how?

I suggest that you read the related SEP entry on the philosophy of mathematics.

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