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Each edge of the following cube is 1 and C is a point on the edge.

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What would the height of triangle be in this case , how would you measure it?

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Assuming you mean "height" as "altitude passing through point $C$". You might want to clarify if this is not correct:

The height of the triangle is just the distance from $C$ to the line $AB$. Cutting the cube perpendicular to $AB$ and containing point $C$, we find a right angled triangle with both legs of length $1$, and the height of the triangle as its hypotenuse. So the height of the triangle is $\sqrt{1^2+1^2}=\sqrt 2$

It's also good to realize that the height doesn't change depending on where $C$ is, so you can just arbitrarily move it to one vertex and take the height along the corresponding face.

Picture:

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    I'm learning how to use Geogebra anyway; it was my pleasure.2012-06-11