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.