The necessity for compact table-top x-ray sources with higher brightness, shorter wavelength and shorter pulse duration has led to the development of complementary sources based on laser-plasma accelerators, in contrast to conventional accelerators. Relativistic interaction of short-pulse lasers with underdense plasmas results in acceleration of electrons and in consequence in the emission of spatially coherent radiation, which is known in the literature as betatron radiation. In this article, we report on our recent results in the rapidly developing field of secondary x-ray radiation generated by high-energy electron pulses. The betatron radiation is characterized with a novel setup allowing to measure the energy, the spatial energy distribution in the far-field of the beam and the source size in a single laser shot. Furthermore, the polarization state is measured for each laser shot. In this way, the emitted betatron x-rays can be used as a non-invasive diagnostic tool to retrieve very subtle information of the electron dynamics within the plasma wave. Parallel to the experimental work, 3D particle-in-cell simulations were performed, proved to be in good agreement with the experimental results.