What is symplectic geometry? EDIT: Much thanks for answers. As was pointed out, the question as it stands is a little too broad. Nevertheless, I don't want to delete it, because I think that such introduction-style questions can be answered without writing a book, rather something more like an introduction to a book and fits here. Moreover, commenters have linked to great resources, and this question might help someone else. I made a follow up strictly narrower question instead.

First some background, so that you know where I came from. But the question in the title stands as it is, if you want to answer without appealing to what is below, please do.
I am currently learning about Lie groups. One of the first things that I've seen are the classical groups, and the classical group that I want to talk about today is the symplectic group $\mathrm{Sp}(n,\mathbb{F})$.
The definition of $\mathrm{Sp}(n,\mathbb{F})$ I am familiar with is as follows:
Let $\omega$ be an skew-symmetric bilinear form on $\mathbb{F}^{2n}$, which is unique up to change of basis. It is given by the formula $$\omega(\mathbf{x},\mathbf{y}) = \sum_{i=1}^n{x_iy_{i+n}-y_ix_{i+n}}$$
Why is this symplectic form important?
We can then write out the definition
$$\mathrm{Sp}(n,\mathbb{F}) = \left\{ A: \mathbb{F}^{2n} \to \mathbb{F}^{2n} \mid \omega(A\mathbf{x},A\mathbf{y}) = \omega(\mathbf{x},\mathbf{y}) \text{ for all } \mathbf{x,y} \in \mathbb{F}^{2n}\right\}$$
I can see the analogue of $O(n,\mathbb{F})$. We also have some bilinear form that needs to be preserved, namely the inner product $\langle \cdot,\cdot\rangle$. But more importantly, elements of $O(n,\mathbb{F})$ are really easy to visualize, because I intuitively know what a rigid transformation is. So the important question for me is
How to visualize symplectic transformations?
And I tried to research this question, and I stumbled upon the topic of symplectic linear spaces and symplectic manifolds. A symplectic vector space is defined analogous to Euclidean vector space, but the inner product is again substituted by symplectic form.
What is a symplectic vector space, intuitively?
I saw that the intuition behind these things should be that $\mathbb{R}^{2n}$ should be treated as a space of positions and velocities, a phase space. And I don't understand it. But I feel that physical intuition would be really helpful.
What is the connection of classical mechanics with symplectic geometry?
I don't know classical mechanics, sadly, so a quick mathematical rundown would be appreciated.
All the questions that I've asked above could be summarized to one question:
What is symplectic geometry?
 A: Quick "fake" answer: In classical mechanics one usually describes a particle measuring its position $q_1, \dots, q_n$ and momentum $p_1, \dots, p_n$. To describe how these change one needs to introduce a "Hamiltonian", i.e. a function measuring the energy of the system. 
For a particle of mass $m$ moving in the ordinary space $\mathbb R^n$ it is:
$$H(q, p) = \frac{p_1^2 + \dots + p_n^2}{2m} + V(q)$$
where $V\colon \mathbb R^n\to\mathbb R$ is the "potential energy" of the particle. Then one solves a system of ODEs:
$$\begin{cases} \dot p_i = -\frac{\partial H}{\partial q_i} \\ \dot q_i = \frac{\partial H}{\partial p_i} \end{cases}$$
For example if you plug $n=1$ and $V(q) = kq^2/2$, you will get an ordinary harmonic oscillator $q(t)=A\cos(\omega t+\phi)$, $\omega^2=k/m$. (Similarly you get an expression for the momentum $p$).
Now let's generalize. One starts with a configuration space that is a manifold $M$, used to measure the position of the particle. Local coordinates are our $q_1, \dots, q_n$.
Then one introduces the phase space $P=T^*M$ on which the local coordinates are $q_1, \dots, q_n, p_1, \dots, p_n$. The motion of the particle can be described by a path on $P$, which measures not only the position but the momentum as well. We do this by introducing a function $H\colon P\to \mathbb R$ and we try to find a vector field on $P$ such that:
$$i_X\omega=-dH,$$
where $\omega = dp_1 \wedge dq_1 + \dots + dp_n\wedge dq_n$ in local coordinates. (It is not obvious that it is globally defined). This (not incidentally) looks similar to the expression $\omega(\textbf x, \textbf y)$ you have written down in the question.
The point is that the whole dynamics is in fact encoded in the symplectic 2-form $\omega$. (If you have a Hamiltonian describing a particle, just find a vector field and solve an ODE to get the path).
Generalizing even further let's think about a symplectic manifold $(P, \omega)$ where $\omega$ is a distinguished 2-form with 'nice' properties (it's assumed to be closed and nondegenerate). In particular this gives some topological restrictions on $P$ – for example $P$ needs to be even-dimensional and orientable, with $\omega\wedge \dots\wedge \omega$ acting as a volume form.
Obviously one can organize such manifolds into a category and ask the usual questions – can we characterize them up to an isomorphism? (Called 'symplectomorphism'; strongly related to 'canonical transformations' of physics). Can we introduce any invariants? (Apparently there are no local ones as every symplectic manifold locally looks like $\mathbb R^{2n}$ with the symplectic form from your question).
As we can do classical mechanics on such manifolds, can we 'quantize' them and do quantum mechanics? 
We have a nice additional structure – how does it interfere with a Riemannian metric or complex structure (what leads to Kähler geometry and Calabi-Yau manifolds of string theory).
... and similar questions seem to be so ubiquitous that I'd risk to say: every modern differential geometer needs to learn symplectic geometry.
Full answer: This is too broad subject to describe it fully here. But definitely it's worth to study. I recommend: 


*

*Cohn's post,

*Webster's post,

*Cannas da Silva's notes,

*Meinrenken's notes,

*Butterfield's On Symplectic Reduction in Classical Mechanics,

*Arnold's Mathematical Methods of Classical Mechanics,

*Abraham and Marsden's Foundations of Mechanics,

*McDuff and Salamon's Introduction to Symplectic Topology.

