Normal coordinates, normal form, canonical form, canonical coordinates etc are all overused terms. But roughly speaking, the intended meaning in each case is something like:
thing 1 "looks like" thing 2, and thing 2 is a 'simple' or 'standard' or 'well known' object.
The meaning is very context-specific. For example:
- Riemann-normal coordinates
- normal modes/coordinates when discussing oscillations
- Jordan normal form or Jordan canonical form of a square matrix
- rational canonical form of a matrix (and amusingly, the meaning of the word "rational" here is not limited to $\Bbb{Q}$, but rather that no field-extensions are required (and here I'm using field in the sense of abstract algebra, not the Physics sense))
- canonical coordinates in Hamiltonian mechanics/symplectic geometry
- Sylvester normal/canonical form of a bilinear functional, in the sense of Sylvester's "law of inertia" (but of course this isn't really about inertia in the sense Newton meant, but rather more colloquially that it is something which 'resists change' under a suitable change of basis).
Although in each case here we're using the same words normal or canonical, the meaning is different in each case. We have a finite number of words, so we have to overuse them. Thus, you should not rely on words alone to decipher the meaning of terms.
Anyway for your question about normal coordinates: given any smooth manifold with a connection $(M,\nabla)$ (regardless of whether or not $\nabla$ is torsion-free or whether or not it is the Levi-Civita connection of some semi-Riemannian metric $g$), and any point $p\in M$, one can always find a coordinate system $(U,\phi)$ around $p$ such that the Christoffel symbols relative to this system all vanish at $p$ (but not necessarily on all of $U$). Vanishing of Christoffel symbols in a neighbourhood of each point is equivalent to the connection having zero curvature (a very strong condition).
Note that existence of such normal coordinates is simply a differential-geometric statement about connections, and it's a very general fact: not specific to metric tensors, so nothing specific to GR, let alone linearized gravity. So, it also has nothing, a-priori, to do with oscillations. The nice thing about normal coordinates is that a good understanding of them helps with analyzing properties of auto-parallel curves (i.e curves $\gamma$ satisfying the equation $\nabla_{\dot{\gamma}}\dot{\gamma}=0$); which are of course a very special and interesting class of curves to study (in the metric case, these are called the geodesics).
