A pulsar wind nebula (PWN) is a system with a central rotating object, a pulsar, which powers a surrounding nebula. This complex system is observed to emit radiation throughout the electromagnetic spectrum, from radio to gamma rays. In this thesis I explored the gamma-ray emission from pulsars, their respective nebulae and one PWN candidate at very high energies (VHE, E > 100 GeV) using the Major Atmospheric Gamma-ray Imaging Cherenkov (MAGIC) telescopes. At the beginning of this research, only one pulsar, the Crab pulsar, was known to emit at VHE, challenging the theoretical models. The observed electromagnetic radiation from the pulsar-nebula system implies the presence of a mechanism that accelerates charged particles to ultra-relativistic energies. However, this mechanism is poorly understood, thus VHE pulsar observations are relevant not only as information for emission modelling but also as a contribution for better characterization of fundamental properties of these complex astrophysical systems. The MAGIC telescopes, with its novel trigger especially developed for pulsar observation, is the most suitable instrument to search for new VHE pulsars. In this thesis, I present the observations and analysis of two galactic sources: Crab (PSR J0534+2200) and Dragonfly (PSR J2021+3651) pulsars. For the well-known and previously detected Crab pulsar, the analysis shows the expected results and is used as a performance check of the method. The same method is then used to search for VHE emission from the Dragonfly pulsar, proposed as a very likely VHE pulsar candidate due to its similar characteristics to the Crab pulsar. I found no significant pulsed emission from Dragonfly pulsar in low E -range (50 GeV < E < 200 GeV) neither in the full E -range (E > 200 GeV), only upper limits were derived. However, with the same set of data, the Dragonfly nebula surrounding the pulsar was detected. I also present the study of the unidentified TeV source, HESS J1858+020, that was put forward as a relic PWN candidate using the archival data collected from the MAGIC telescopes where this source was relatively far from the centre of the camera implying decrease in detection sensitivity. Nevertheless, the source was detected, source extension estimated and the spectrum between 300 GeV and 10 TeV was constrained, but morphological or other details were not discerned. Neither the PWN scenario could be refuted nor confirmed. Overall, detail studies of the VHE gamma-ray emission from pulsars and nebulae seem more challenging than expected, and longer observations are needed for pulsed detection or for morphological characterization of nebulae. The next-generation Cherenkov telescope array (CTA) with an order of magnitude better sensitivity and with 1 arcminute resolution, will certainly allow detail morphological and spectral studies of this kind of sources. Furthermore, I also studied a faint component of the Galactic diffuse synchrotron emission at low-radio frequencies by using multiple polarimetric observations with the LOw Frequency ARray (LOFAR). Before stacking them, these observations first needed to be corrected for the Faraday rotation in the Earth’s ionosphere, otherwise the observed polarized emission may be either partially or in exceptional cases fully depolarized. I used the observed polarized diffuse synchrotron emission to characterize and additionally to correct for the ionospheric Faraday rotation. After stacking twenty observations, the noise was reduced by ~ √20, as expected. Higher signal-to-noise ratio achieved with this method, enables a study of the faint component of the Galactic diffuse emission, which was not visible in a single reference observation. Moreover, applied technique can also be used for studies of the faint polarized sources, including pulsars.