This is related to the definition of vectors related to a given trajectory ##\vec{x}(t)## in usual 3D Euclidean space. It defines specific (usually non-inertial) reference frame for an observer moving along this given curve.
You start with the normalized tangent vector,
$$\vec{T}=\frac{\dot{\vec{x}}}{|\dot{\vec{x}}|}.$$
For the following it is more convenient to introduce the arc-length parameter,
$$s=\int_{t_0}^t \mathrm{d} t' |\dot{\vec{x}}(t')|,$$
which is the arclength of the piece of curve given by ##t' \in [t_0,t]##.
Obviously
$$\dot{s} =|\dot{\vec{x}}|>0,$$
and thus you can parametrize the curve as well by ##s## rather than ##t##. Then the normalized tangent vector reads
$$\vec{T}=\frac{\dot{\vec{x}}}{\dot{s}}=\frac{\mathrm{d} \vec{x}}{\mathrm{d} s}.$$
Then, because of ##\vec{T}^2=1=\text{const}## you have
$$\vec{T} \cdot \mathrm{d}_s \vec{T}=0,$$
and thus you can define
$$\vec{N}=\frac{\mathrm{d}_s \vec{T}}{|\mathrm{d}_s \vec{T}|}$$
as one unit-normal vector to the curve and finally, as a third vector making a right-handed Cartesian coordinate system complete, the unit-binormal vector
$$\vec{B}=\vec{T} \times \vec{N}.$$
Further you have by definition of ##N##
$$\mathrm{d}_s \vec{T}=\kappa \vec{N},$$
where ##\kappa = |\mathrm{d}_s \vec{T}|##. Now from ##\vec{N}^2=1## you find that
$$\vec{N} \cdot \mathrm{d}_s \vec{N}=0 \; \Rightarrow \; \mathrm{d}_s \vec{N}=\alpha \vec{T} + \tau \vec{B}$$
and then
$$\mathrm{d}_s \vec{B} = (\mathrm{d}_s \vec{T}) \times \vec{N} + \vec{T} \times \mathrm{d}_s \vec{N} = \kappa \vec{N} \times \vec{N} + \vec{T} \times (\alpha \vec{T}+\tau \vec{B}) = \beta \vec{T} \times \vec{B} = \tau \vec{T} \times (\vec{T} \times \vec{N}) = \tau [ \vec{T} (\vec{T} \cdot \vec{N})-\vec{N} \vec{T}^2]=-\tau \vec{N}.$$
In matrix notation we can write
$$\mathrm{d}_s (\vec{T},\vec{N},\vec{B}) = (\vec{T},\vec{N},\vec{B}) \begin{pmatrix} 0 & \alpha & 0 \\
\kappa &0 & -\tau \\ 0 &\tau &0 \end{pmatrix}=(\vec{T},\vec{N},\vec{B}) \hat{M}. \qquad (*)$$
But now the matrix
$$\hat{O}=(\vec{T},\vec{N},\vec{B}),$$
built with the column vectors ##\vec{T}##, ##\vec{N}##, and ##\vec{B}## is an orthogonal matrix, because these three vectors form a right-handed Cartesian basis, i.e.,
$$\hat{O} \hat{O}^{\text{T}}=\hat{1}$$
and thus
$$\mathrm{d}_s (\hat{O} \hat{O}^{\text{T}})=0=(\mathrm{d}_s \hat{O}) \hat{O}^{\text{T}} + \hat{O} \mathrm{d}_s \hat{O}^{\text{T}},$$
which means that the matrix ##\mathrm{d}_s \hat{O}) \hat{O}^{\text{T}}## is antisymmetric. Now we can write (*) as
$$\mathrm{d}_s \hat{O} = \hat{O} \hat{M} \; \Rightarrow \; \hat{M}=\hat{O}^{\text{T}} \mathrm{d}_s \hat{O},$$
which implies that ##\hat{M}=-\hat{M}^{\text{T}}## and thus ##\alpha=-\kappa##, i.e., finally
$$\mathrm{d}_s \vec{T}=\kappa \vec{N}, \quad \mathrm{d}_s \vec{N} = -\kappa \vec{T} + \tau \vec{N}, \quad \mathrm{d}_s \vec{B}=-\tau \vec{N}.$$
These are the Frenet-Serret formulas, and ##\kappa## and ##\tau## are called curvature and torsion of the curve.
The geometrical meaning becomes clear from the following figure from Wikipedia:
https://commons.wikimedia.org/wiki/File:Frenet.svg#/media/File:Frenet.svg