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In the Wikipedia article about Bell states it is written:
Independent measurements made on two qubits that are entangled in Bell states positively correlate perfectly, if each qubit is measured in the relevant basis.
What does it even mean to perform a measurement in a certain basis?
You can answer by using the example of the Bell states of the Wikipedia article.
Qubits are essentially quantum objects from which you can extract a bit. But there are different ways that this can be done, and the answer you get depends on the measurement you choose.
If you qubit is an electron spin, the measurement basis corresponds to measuring spin in a particular direction. We use that picture more generally in the form of the Bloch sphere. Measurement in this case corresponds to taking pair of opposing points on the sphere and making the qubit choose between then. Each possible pair of opposing points is referred to as a different measurement basis.
Often with qubits, practical reasons in implementation mean that we can only actually measure in a single basis, known as the $Z$ or computational basis. To simulate the others we can precede our measurement with a certain single qubit rotation. The rotation we choose determines the basis we end up measuring in.
For a given Bell state, and for a given measurement basis on one of the qubits, there exists a measurement basis for the second with with results will be perfectly correlated. This seems to be what the article is getting at.
For the Bell states, measuring both qubits in the $Z$ basis will either end up with perfect correlation or perfect anti correlation, depending on which Bell state it is. If you get an anti correlation, you can change basis on one qubit by applying an $X$ rotation before the measurement. This new basis will get perfect correlations with the $Z$ basis measurement results for the other qubit.
If you think of a electronic spin $S=1/2$, imagine measuring it on the z-axis to obtain $S_z=+1/2$ (or $S_z=-1/2$). This (the z projection of the spin magnetic moment) is a possible basis for the measurement. Or you could measure the spin on the x-axis, and they you will obtain $S_x=+1/2$ (or $S_x=-1/2$). This is a different basis.
The measurements on Bell pairs will correlate with each other when measured on non-orthogonal basis (if you measure one particle on z and the other on x, the results will be perfectly uncorrelated; if you measure both on z or both on x, the results will be perfectly correlated).
Other examples of measurement bases would be polarization with photons: vertical vs horizontal is the linear polarization basis, whereas clockwise vs anticlockwise is the circular polarization basis.
What does it even mean to perform a measurement in a certain basis?
It is very close to a measurement of a certain observable. In quantum mechanics, when we talk about measuring an observable, we usually are primarily interested in an eigenvalue as an outcome of the measurement. In quantum information, we don't care about the eigenvalues; we are solely interested in a state after the measurement, and this state can be interpreted as an eigenvector of an observable being measured.
Mathematically, for any "measurement in a certain basis" there exists many observables that correspond to the same measurement (not all of them have physical meaning); all these observables have the same eigenvectors (which form the measurement basis) but may differ in eigenvalues. Eigenvalues don't matter provided they are different, so the measurement distinguishes between the eigenvectors (measurement basis states).
