Rigorous canonical coordinate definition I would like to know what is the definition of a canonical coordinate.
Let's assume we have a Lagrangian $\mathcal{L}(q,\dot{q})$. Are the canonical coordinate simply the set $(q,p)$ where $p=\frac{\partial \mathcal{L}}{\partial \dot{q}}$?
Are the canonical coordinates related to canonical transformations? I.e. if I start with some canonical coordinates and I do a canonical transformation I will still have canonical coordinates?
The point of my question is because I need to compute some Poisson bracket and I would like to be sure that I can use the $(q,p)$ obtained from the initial Lagrangian as canonical coordinates.
I saw on wikipedia https://en.wikipedia.org/wiki/Canonical_coordinates

Canonical coordinates can be obtained from the generalized coordinates of the Lagrangian formalism by a Legendre transformation, or from another set of canonical coordinates by a canonical transformation.

But as it is really briefly said and I didn't find any confirmation of this elsewhere I would like to be absolutely sure.
 A: *

*Definition. Given a $2n$-dimensional symplectic manifold $(M,\omega)$ with an open neighborhood $U\subseteq M$, a local coordinate system 
$$(q^1,\ldots,q^n,p_1,\ldots,p_n): U\to\mathbb{R}^{2n}\tag{1}$$ 
is called canonical coordinates/Darboux coordinates if the restriction of the symplectic 2-form $\omega$ takes the form
$$  \omega|_{U}~=~\sum_{j=1}^n\mathrm{d}p_j\wedge\mathrm{d}q^j.\tag{2} $$
Eq. (2) is equivalent to the fundamental Poisson bracket relations.

*In classical mechanics, the main example of a symplectic manifold is the cotangent bundle of the configuration space equipped with the tautological one-form.
A: Elaborating on Qmechanic's answer in a "less mathematical" manner.
A symplectic space (manifold really, but apparantly OP prefers it without manifolds) is a coordinate space, in which there is an antisymmetric tensor field $\omega_{\mu\nu}$ that satisfies two properties:
1) closed: $\partial_\kappa\omega_{\mu\nu}+\partial_\mu\omega_{\nu\kappa}+\partial_\nu\omega_{\kappa\mu}=0$;
2) nondegenerate: $\det\omega\neq 0$.

The fundamental structure of Hamilton's mechanics is that of a symplectic space. In elementary treatments, this aspect is usually not emphasized. Relating the usual Poisson structure to the symplectic structure can be accomplised by writing for any two phase space functions $f,g$ $$ \{f,g\}=B^{\mu\nu}\partial_\mu f\partial_\nu g. $$ Here the greek indices $\mu,\nu,...$ run through all phase space coordinates (position and momentum). Then $\omega_{\mu\nu}$ is the inverse of $B^{\mu\nu}$.

The following theorem is due to Darboux (or at least is called Darboux' theorem):
If we define $(J_{\mu\nu})=\left(\begin{matrix} 0 & \mathbb I_n \\ -\mathbb I_n & 0\end{matrix}\right)$ ($2n\times 2n$ block matrix of size $n\times n$ blocks, maybe the signs are opposite, doesn't matter that much), then there exists local coordinates $(x^\mu)=(q^i,p_i)$ (latin indices run $1,...,n$, greek $1,...,2n$) such that in those coordinates $$ \omega_{\mu\nu}=J_{\mu\nu}. $$
Such coordinates are called Darboux coordinates in the mathematical literature and canonical coordinates in the physics literature.
Canonical transformations are those coordinate transformations that map canonical coordinates to canonical coordinates.

Hamiltonian mechanics is independent of Lagrangian mechanics in the sense that one does not have to necessarily begin with a Lagrangian description. What is needed to do Hamiltnian mechanics is a symplectic space and a Hamilonian function.
However if one starts off with (nondegenerate) Lagrangian mechanics, then the map $(q,\dot q)\mapsto (q,p),\ p=\partial L/\partial \dot q $ will always produce canonical coordinates. Not all canonical coordinate systems however arise this way.
