Question

What is the significance of a particle being in a mixed state when quantum mechanical measurements are being made on it?

Answer 1

Wikipedia or Google Schrodinger's Cat. The simple answer to your question is that taking a measurement of such a particle necessarily forces it into one state that is not its "natural" one. The complicated answer includes things like the Heisenberg Uncertainty Principle (also worth looking into), a related concept stating that there is a limit to how precisely two related variables in a system can be measured simultaneously, and the resulting uncertainty is greater as the particle in question gets smaller.