How and under what principle are measurements made at quantum level? There's a lot of news about amazing quantum phenomena, but even without the expert knowledge I would still like to know how it is done and think about the possibility of errors in measurements.
How, in other words using what physical method can quantum level measurements be made? (Basic enough so most people can understand.)
And using what scientific principals is this method based on?
 A: A quantum measurement is always the macroscopic readout of a macroscopic experimental setup. Particle detectors are usually converting the energy of a particle (be that a photon, electron, proton etc.) into an electronic signal that represents the position and/or momentum of the particle. Spins can be converted into photon signals (or, in case of many spins at once, electromagnetic waves), that can, again, be measured with electronic detectors. The actual design of these detectors is a completely different matter than understanding the dynamics of quantum systems. 
So it really depends on what you are asking? Would you like to know how to build detectors that can detect single particles? In that case I could suggest you read some literature about microchannel plates for electrons and photomultiplier tubes for visible photons. Hamamatsu has some nice technical handbooks about their construction and operation here: 
http://www.hamamatsu.com/us/en/support/lib/index.html?spsort=cz&spkey=pmt&screen=lib&spcats=Handbook 
Then there are electronic detectors like CCDs http://en.wikipedia.org/wiki/Charge-coupled_device and even more sensitive bolometers http://en.wikipedia.org/wiki/Bolometer. As we move up the energy scale, we are getting to deal with semiconductor detectors for gamma rays and charged particles that look (and work) a little like CCDs  https://www.stfc.ac.uk/PPD/resources/pdf/lecture02_weber.pdf. Measuring the energy of high energy particles is done in e.g. Argon Calorimeters like this one http://iopscience.iop.org/1742-6596/160/1/012043/pdf/1742-6596_160_1_012043.pdf. Strong magnetic fields are used to distinguish charged particles from uncharged ones, because they bend the charged ones on semi-circular tracks. There are entirely gas based detectors like wire chambers http://en.wikipedia.org/wiki/Wire_chamber and the ingenious Time Projection Chamber http://en.wikipedia.org/wiki/Time_projection_chamber, which are usually surrounded by a strong electromagnet. Multiple of these detector technologies are used together in experiments like ATLAS and CMS (both at LHC). 
Among the largest detectors ever built are Cherenkov detectors like Super Kamiokande http://www-sk.icrr.u-tokyo.ac.jp/sk/detector/howtodetect-e.html and IceCube http://icecube.wisc.edu/. Even larger are cosmic ray arrays like http://www.telescopearray.org/.
The practice of these measurements is an engineering art in itself. What I have given you is just a tiny cross section of methods that we use to make measurements of quantum phenomena across the energy range of infrared light to the highest energies of the universe. 
About spin detection: In ordinary nuclear spin resonance and electronic spin resonance, we need a very large number of nuclear or electron spins to add up to a macroscopic magnetic signal, that can basically be detected by a coil or antenna and a sensitive electronic circuit, that is basically just a specialized "radio". The total energy that we can extract from the sample is basically the number of excited spins times the energy of the spin transition. Since the latter is very small, we need a large number of spins to add up to a measurable signal. The limit of signal that we can measure is given by the input noise of our "radio", which is dominated by thermodynamic noise that is many orders of magnitude larger than a signal from a single spin.  
Optically detected spin resonance is much more sensitive than direct detection, theoretically we can detect single atoms, because we are using the atoms themselves as noiseless amplifiers. The basic principle couples an optical transition between two atomic or molecular states, which have an energy difference that corresponds to a visible photon, to a spin transition, that has a very small energy gap. 
A typical experiment will prepare the atoms in a spin state using a magnetic field. Then a laser is used to "pump" the atom to the upper optical state. Now we have the required energy to make the atom give off a photon. Because of the coupling between the spin state and the optical transition, the probability, that the atom will emit a fluorescence photon is now modulated by the very weak spin state. 
A typical experimental setup is shown in http://portal.tugraz.at/portal/page/portal/TU_Graz/Einrichtungen/Institute/Homepages/i5110/forschung/heliumnanodroplets/Electron%20spin%20resonance. Here the experiment loads tiny helium droplets with potassium or rubidium atoms. Elements like potassium and rubidium have a suitable coupling between optical and magnetic states. When they are exposed to a magnetic field, the energy levels of the spin states split (spins parallel to the magnetic field have a different energy than spins which are antiparallel). The first laser source pumps the optical energy into the atoms. The spin of the excited atoms can now be changed with a microwave signal (in this case, 9.4GHz). A second laser (or sometimes just spontaneous emission), then reads out the atomic state by making the atom return into its electronic ground state by emitting a photon. Since that emission is usually detected perpendicular to the laser light (and since one can introduce a time delay between the laser and the detected signal), a sensitive detector like a photomultiplier tube can be used to pick up single photons. 
In any case, even though the quantum mechanical system is tiny, the detector is macroscopic. A typical photomultiplier tube used for these experiments is half an inch to an inch in diameter, much, much larger than the atoms that send out the photons. 
