Bell state
A Bell state is a two-qubit state with maximal entanglement, sort of like two flipping coins that are 100% correlated. For example, two photons produced by parametric down-conversion are a Bell pair, and together the pair are in a Bell state.
Bell basis
The Concept of Basis Set
When you measure two qubits, you can get four possible measurements, so any 2-qubit state is described by a 4-vector. Any complete basis set for this space will have four basis vectors.
The Calculation Basis Set
In the calculation basis, these four basis vectors can each be written as integer kets, binary kets, or vectors:
$|0\rangle=|00\rangle= \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}$,
$|1\rangle=|01\rangle= \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}$,
$|2\rangle=|10\rangle= \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}$, and
$|3\rangle=|11\rangle= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$.
Although that particular basis set has four basis vectors that are unentangled states, there are infinitely many basis sets.
The Bell Basis Set
In the Bell basis, the basis vectors (the four Bell states) are each maximally entangled:
|$\Psi^\pm\rangle=\frac{1}{\sqrt 2}(|00\rangle\pm|11\rangle)=\frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 0 \\ 0 \\ \pm1 \end{bmatrix}$ and
|$\Phi^\pm\rangle=\frac{1}{\sqrt 2}(|01\rangle\pm|10\rangle)=\frac{1}{\sqrt{2}} \begin{bmatrix} 0\\1 \\ \pm1 \\0\end{bmatrix}$.
Bell measurement
Quantum Measurement
Any two-qubit state can be expressed in any basis. A measurement in a particular basis has eigenvectors in that set, and a measurement will observe the eigenvalue of one of the basis vectors and leave the system in that eigenvector. The squared amplitudes of the basis vectors give the probabilities of measuring that basis vector's eigenvalue.
Measurement in the z Basis
For example if the state
$|\psi\rangle = 0.6 |0\rangle + 0.8 |3\rangle = \begin{bmatrix}0.6\\0 \\0 \\0.8 \end{bmatrix}$
is measured in the calculation basis, there is a $0.6^2$=36% probability of measuring 0 and 64% probability of measuring 3.
Measurement in the Bell Basis
That same state can be expressed in the Bell basis:
$|\phi\rangle = 0.6\cdot\frac{1}{\sqrt 2}(\Psi^++\Psi^-) + 0.8\cdot\frac{1}{\sqrt 2}(\Psi^+-\Psi^-) = 0.99\Psi^+ + 0.14\Psi^-$ with probabilities of 98% and 2% to measure $\Psi^+$ and $\Psi^-$ respectively
In practice
The usual way to measure in the Bell basis is to transform the basis set to calculation basis. Because Hadamard and CNOT are their own inverses, the four Bell states map to the four calculation basis states if H is applied to the conditional qubit after a CNOT is applied.
If the two qubit to be projected onto a Bell state are not in the same location, CNOT is not practical, but an ebit (the entanglement resource of a Bell pair, distributed to the two locations) can be consumed to make the measurement following https://arxiv.org/abs/1612.08578.