What has been measured at the Higgs experiment and what do we know now? Explained at the level of a 5$^{\text {th}}$ semester physics student (i.e. pre QFT, but far beyond the level of a news article for non-physicists, which avoids all details and only deals in analogies) ...


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*What has been measured at CERN some days ago?

*What are the essential ingredients of the theory necessary to interpret said measurement? And so how do we deduce from the results that there is a new field/particle observed?

*How to read the most relevant graphs in the presentation of the results?

 A: 
What has been measured at CERN some days ago?

Once you have an event (as @user1504 describes in his answer), i.e. a proton proton interaction, and have the four vectors of all the interaction products, i.e. jets, photons, leptons, you can generate invariant masses of the interaction products. The Higgs particle has been predicted  theoretically and is the last piece in the jig saw puzzle that the Standard Model has put together in one picture. The invariant mass of two photons shows an enhancement consistent with the Higgs properties. Also the invariant masses of Z Z to a lesser extent. The combined statistics of all possible decay channels of the Higgs seen give a 5sigma certainty that the resonance is there and has the expected decay behavior within statistics of the individual decay channels.

What are the essential ingredients of the theory necessary to interpret said measurement? And so how do we deduce from the results that there is a new field/particle observed?

The theoretical part is a different question and should be posed independently: why and how the standard model theory accommodates all particle data.
That there is a new particle observed with 5 sigma certainty is what the data tells us, and it is new because we have never observed a 125GeV resonance before the present experiments.
To gauge whether it is  the expected from the SM theory Higgs particle one needs good statistics in all the decay channels plus angular distributions that will establish the spin parity. Th theoretical Higgs has 0 spin and positive parity.

How to read the most relevant graphs in the presentation of the results?

The resonance is clear in fig 3. The sigma of this channel is its statistical significance for it alone. Similar for the ZZ channel, fig 4.
Fig 5 gives the probability,

The observed probability (local p-value) that the background-only hypothesis would yield the same or more events as are seen in the CMS data, as a function of the SM Higgs boson mass for the five channels considered. The solid black line shows the combined local p-value for all channels.

A: You've asked for a lot here, so I'm only going to give a partial answer.  Hopefully other people will add other details.
I want to talk about the quantities we measure with particle detectors.
Particle experiments deal in the effective physics of quantum fields at high energy and low density.  Most of the experiments at CERN measure the energy and/or momentum deposited by a particle within a certain volume of space around the expected collision point.  A hadron calorimeter for example is basically a big hunk of metal surrounded by photon counters.  (For other examples, see the lovely wikipedia pages on ATLAS & CMS.)  Strongly interacting particles coming out of the collision smash into the electrons & nucleons in the metal, accelerating them, which causes them to radiate photons.  The photons are detected by the photon counters, and you can work out from the photon arrival counts what sort of particles could have crashed into nuclei and where (to within a small fraction of a steradian).   From the mass, angle, and energy you can reconstruct the 4-momentum, and from the collection of particle types and 4-momentum, you can work out the cross section and branching ratios empirically.
You can also extract this stuff from a QFT model.  Whether you use perturbation theory or monte carlo simulations or old school nuclear physics depends on exactly which observable you've got data on.  For the Higgs measurements that have been in the news, it's mostly perturbation theory to get to the empirical predictions from the theory.  (However, the empirical measurements rely heavily for calibration on numerical computations done in situations where perturbation theory is difficult.)
