This is just a choice of basis of vector fields in a smooth manifold really. Let $(M,g)$ be a $D$-dimensional smooth manifold with Lorentzian metric. Let $U\subset M$ be one open set and let $x:U\to \mathbb{R}^D$ one local chart. Inside $U$ we have a basis of vector fields, the coordinate frame $\partial_\mu$. They allow you to expand a general vector field $X:M\to TM$, inside that open set, as $$X|_U = X^\mu \partial_\mu\tag{1}.$$
Now, this basis is not necessarily orthonormal. In fact the inner products between basis vectors gives you the metric components in the associate coframe basis, $$g_{\mu\nu}(x)=g\left(\partial_\mu,\partial_\nu\right),\quad x\in U\tag{2}$$
and it is clear that $g_{\mu\nu}\neq \eta_{\mu\nu}$ in general. We can, however, always find one orthonormal basis locally. Let us suppose we have found such a basis in another open set $U'\subset M$, and let us denote such a basis by $e_a$ where $a=0,\dots, D-1$ identifies the different vectors. Since each $e_a$ is just a vector, in the overlap $U\cap U'$, which we shall assume non-empty, we may expand each $e_a$ the coordinate basis: $$e_a = e_a^\mu \partial_\mu.\tag{3}$$
This defines a $D\times D$ matrix $e_a^\mu$ which contains the components of the vectors making up the orthonormal frame. Such orthonormal frame is what in Physics one often calls a vielbein.
Now consider the vector field $X$ again. Inside $U\cap U'$ we have two bases available: the coordinate basis $\partial_\mu$ and the vielbein $e_a$. We can expand $$X = X^\mu \partial_\mu = X^a e_a.\tag{4}$$
Denoting the inverse matrix of $e_a^\mu$ by $e^a_\mu$ you can easily check that the relation between components (4) implies is $X^a = e^a_\mu X^\mu$. For a general vector field both sets of components will depend on the manifold point you evaluate it, i.e., it will in general be non-constant.
There are two main uses for this:
As usual, working with one orthonormal basis often gives great simplifications. This is something that should be familiar from linear algebra and all of Physics really.
A deeper reason to introduce this construction is to be able to define the concept of spin structures and introduce spinor fields. It is not possible to give a self-contained discussion of this subject in this answer, but the basic idea is that $(M,g)$ has one natural ${\rm SO}(1,d-1)$ bundle defined over it, the bundle of orthonormal frames. A choice of vielbein $e_a$ on an open subset trivializes this bundle over the open set. If you choose an open covering of $M$ and define vielbeins on all of them, the way the vielbeins are matched on the overlaps basically define the structure of the bundle.
Now depending on the topology of $M$ you may use this construction to build a ${\rm Spin}(1,D-1)$ bundle over $M$ called bundle of spin frames. When this is done, you have one spin structure and you can define spinor fields over $M$. So, in a sense, to be able to introduce spinor fields in a general, possibly curved Lorentzian manifold $(M,g)$, the first step is to introduce vielbeins. A minor comment here is that not all manifolds admit spin structures and that those admiting spin structures might have more than one.
