Here's an extremely boiled-down version of an answer to your question ("What is the measurement process?"). Though your question is a very big question that you should be more specific on, I think this explanation is the best way to get started thinking about these ideas.The measurement process involves two objects: the object being measured and the measurement device. The object being measured is arbitrary; let's call it "the sample." The measurement device can be thought of as the kind you see in everyday experience (a multimeter, oscilloscope, photomultiplier tube, etc.)--a macroscopic object which has different "pointer" states; i.e. a macroscopic device which takes on a state corresponding to the result of the measurement performed on the sample, such that a human looking at the device can tell what state it's in. In the spirit of quantum mechanics, we could write the measurement device's state as a quantum state, just like any other quantum system.
Now let's focus on a specific example to make everything concrete. Let's say that the subject is an electron, and suppose we have a measurement device which measures the electron's spin. We will write the electron's state \left |\psi \right \rangle_e as a linear combination of up and down states (with respect to the z-axis): \left |\psi \right \rangle_e = a\left |\uparrow \right \rangle_e+b\left |\downarrow \right \rangle_e.
Now the measurement device, described by a state \left |\phi \right \rangle_m is one that starts out in some neutral state \left |\phi \right \rangle_m = \left |0 \right \rangle_mand then changes its state indicating whether the electron was spin up or spin down--if it sees the electron as spin up, the measurement device's state becomes \left |\phi \right \rangle_m = \left |\uparrow \right \rangle_m, whereas if it sees the electron as spin down, the measurement device's state becomes \left |\phi \right \rangle_m = \left |\downarrow \right \rangle_mAs an example, let's write down how the measurement device would measure an electron which happens to be in the spin up state: Before the measurement, the electron-measurement device system is in the state
\left |\psi \right \rangle_e\left |\phi \right \rangle_m = \left |\uparrow \right \rangle_e\left |0 \right \rangle_m
After the measurement is made, the measurement device reflects the electron's state.
\left |\psi \right \rangle_e\left |\phi \right \rangle_m = \left |\uparrow \right \rangle_e\left |\uparrow \right \rangle_m
This particular measurement process can be written in summary as:
\left |\uparrow \right \rangle_e\left |0 \right \rangle_m \rightarrow \left |\uparrow \right \rangle_e\left |\uparrow \right \rangle_m
Now suppose we can prepare the electron in a state such that it is in a superposition made of equal parts up spin and down spin (relative to the z axis. This can be prepared by measuring the spin along the x-axis first.)
\left |\psi \right \rangle_e = \frac{1}{\sqrt{2}}\left (\left |\uparrow \right \rangle_e+\left |\downarrow \right \rangle_e \right )
What happens when the measurement device tries to measure this state?
According to Schrodinger evolution (forgetting about collapse for the moment), and moreover the principle of superposition, we could easily see the following evolution should happen (absent collapse):
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\left |\psi \right \rangle_e \left |\phi \right \rangle_m = \frac{1}{\sqrt{2}}\left (\left |\uparrow \right \rangle_e+\left |\downarrow \right \rangle_e \right )\left |0 \right \rangle_m\rightarrow\frac{1}{\sqrt{2}}\left (\left |\uparrow \right \rangle_e\left |\uparrow \right \rangle_m+\left |\downarrow \right \rangle_e\left |\downarrow \right \rangle_m \right )
However, in subjective experience, when we look at a measuring device, we never see it in the superposition of states above. All we see is either \left |\uparrow \right \rangle_e\left |\uparrow \right \rangle_m or \left |\downarrow \right \rangle_e\left |\downarrow \right \rangle_m. Somewhere along the lines, one part of the quantum state gets thrown out.
This modeling of the measurement process and the apparent contradiction is due to Von Neumann. Though this argument at first appears to support the objective reality of wavefunction collapse, some deeper reflection on it shows that it's just the same as what would happen in the many-worlds interpretation (no collapse). For that argument, watch Sidney Coleman's lecture "Quantum Physics In Your Face" which is available on the Harvard website.
