They can be used to describe a general rotation in three-dimensional Euclidean space using usually the order "once about the x-axis, once about the y-axis, and once about the z-axis". They are also called nautical rotations.
In aeronautical and aerospace engineering they are often called Euler angles, but this conflicts with existing usage elsewhere, because Tait-Bryan rotations have differences with Euler angles described below.
They are intrinsic rotations and the calculus behind them is similar to the Frenet-Serret formulas.
To perform a rotation in an intrinsical reference frame is equivalent to right-multiply its characteristic matrix (the matrix that has the vector of the reference frame as columns) by the matrix of the rotation
The composition of rotations in the fixed axes is a left-multiplication, because the usage of matrices as operators (left-multiply rotates all that is at the left of the operator). Suppose that you write the characteristic matrix of the frame as product of fixed axes rotations. Then, let x(φ) and z(φ) denote the rotations of angle φ about the x-axis and z-axis, respectively. In the moving axes description, let Z(φ)=z(φ), X′(φ) be the rotation of angle φ about the once-rotated X-axis, and let Z″(φ) be the rotation of angle φ about the twice-rotated Z-axis. Then:
Therefore rotations in the intrinsical frame can be performed right-multiplying the matrix of the frame, as we wanted to prove. A simpler way to see this is changing the rotation operator in the intrinsical basis to the external basis. To change an operator to a basis given by a matrix P, we have the expresion R'=P−1.R.P and we need its inverse R=P.R'.P−1, and applying this to the rotation matrix we have P'=P.R.P−1.P=P.R
The main difference is that the set of rotations defined by Euler angles (precesion, nutation and intrinsic rotation) are commutative, and this set of rotations is not.
Other difference is that, as rotation axis are not fixed, their position depends on the first rotation. This complicates calculus, but allows us to reach any final position with only two of the three elemental rotations. For example, a satellite could be stabilized with only two reaction wheels. The existing MAP Safehold/CSS (CSS:coarse Sun sensor) controller can work with only two wheels if the system momentum bias is small.
You can visualize why this can be achieved with an example. An aeroplane doesn't need to perform a yaw to turn. It is enough to make a roll. Then the lift on the wings will force a pitch upwards. At the end, the plane will perform a roll in the opposite direction of the previous one to get the wings horizontal. The whole maneouvre is equivalent to a yaw, but only pitch and roll were performed.
Another good example that any final position can be achieved with only two inertial wheels could be the following picture. With a wheel on the z axis and other in the y axis, we can prove that any position can be reached because starting with z over Z we can perform with a single inertial wheel the first and third Euler angles.
Tait-Bryan rotations, as any other intrinsic rotation, can be composed with each other with no limit. Nevertheless, some compositions of three of them are equivalent to the Euler angles, and share with them their properties. Euler angles therefore can be considered a particular application of Tait-Bryan rotations when the moving frame initial position is the same as the external reference frame and the order in which the rotations are applied is the proper one (if xyz are the reference frame and XYZ the moving frame, the first rotation (yaw) around Z leaves the line of nodes over y, so that the rotation around y (pitch) may be taken as equivalent to rotation about N)
As in a moving frame all these things are true only for an instant, we can only assert them in the limit when time goes to zero. Thus, in a frame co-moving with the rotating system, Euler angles are equivalent to a special combination of Tait-Bryan angles in the limit when delta time goes to zero,
The main usage is in a part of flight dynamics, called attitude control, because the three angles can be controlled separately. If we correct small errors in yaw, roll and pitch individually, then we have achieved the nominal attitude of the aircraft. In case of a unmanned spacecraft, this can be performed automatically with a gyroscope and a reaction wheel controller in each axis.