An example

Let M = S^2 regarded as the set of unit vectors in R3. Tangent vectors

to S^2 at p may then be identified with vectors orthogonal to p. The standard

symplectic form on S2 is induced by the inner and exterior products:

w_p(u, v) := , for u, v in T_{p}(S^^2) =orthogonal {p}.

This form is closed because it is of top degree; it is nondegenerate because is not 0 when u is not 0 and we take, for instance, v = u × p.

Many important spaces come with a 'natural' symplectic form (for example cotangent bundles come with the symplectic form \omega=d\alpha, where \alpha is the canonical 1-form. They are not -naturally- endowed with a Riemannian metric.) Therefore it's a reasonable to try to find the symplectic invariants. As (by Darboux' theorem, and unlike the Riemannian case) any two symplectic forms are locally equivalent, symplectic invariants catch interesting non-local features. Besides volume, 'symplectic capacities' (Hofer, Viterbo) are examples of symplectic invariants. It is not so easy to describe what they measure. One possible description is in terms of the action of periodic orbits of certain Hamiltonian systems defined on the symplectic manifold. This connection between the dynamics of Hamiltonian systems and symplectic geometry is one major reason to find it interesting.

However, it is always possible to chose a Riemannian metric for a given symplectic manifold. The symplectic form can then be expressed with respect to the inner product by means of an almost complex structure. Then, you can try to relate quantities of Riemannian geometry and symplectic geometry. If the almost complex structure is integrable the manifold is 'Kähler', and you and up with 'Kähler geometry'.

p.s. In mechanics the tangent bundles of configuration spaces often 'come' with a non-degenerate quadratic form on the fibres, the kinetic energy, which would allow to define a Riemannian metric on T*M (after some work, via a connection). So for mechanics, it may seem natural to study the symplectic structure _and_ the (kinetic) Riemannian structure.

Born, thank you very much. I think Darboux's theorem here is of vital importance. As it states, there's no local invariant quantity except structural two-form in symplectic geometry. This implies any local quantities make no sense up to coordinate transformation.

There are a couple simple things to notice before anything else. Since a symplectic form is antisymmetric one cannot expect to get a notion of length by considering putting a vector in both slots since that is always zero. A straightforward definition of angle based on a symplectic form is also not possible. One can already see how different the subject should be by comparing inner product spaces with symplectic vector spaces. Any vector is perpendicular to itself with respect to a symplectic form.

There is also a certain lack of rigidity in symplectic geometry because of the Darboux theorem; since all symplectic manifolds are locally "isometric" or locally equivalent, there will not be a local invariant analogous to Riemannian curvature. In Riemannian geometry a manifold is locally isometric to Euclidean space if and only if the curvature tensor vanishes. On the other hand, there are a host of broadly geometric issues and results in symplectic geometry such as Liouville's theorem which involves volume, compatibility with almost complex structures, Kahler metrics and interactions with topology. And, of course, anything concerning symmetry under group operations (or groupsoids, psuedogroups etc.) can be considered geometric in the broad sense and there is certainly a lot of that in the subject; Hamiltonian symmetries, conservation laws/momentum maps, Lie groups acting by symplectic diffeomeorphism, Poisson commuting functions and their flows etc. In short, symplectic geometric is geometric in a much different sense than Riemannian geometry and this is already apparent in the linear theory. While angle and length are not natural notions in the theory, volume does play a major role.

In physics the symplectic form comes in mostly in Classical Hamiltonian mechanics.

The symplectic form is used in definining Darboux coordinates aka canonical - aka action angle - aka impulse momentum - aka  P and Q coordinates , i.e. local coordinates such that the symplectic form looks like

                     \omega = \sum_i  dP_i \wedge dQ_i

The symplectic form is used to define the Hamiltonian flow on phase space. For any function f on phase space (usually denoted f(P_1, ... P_n, Q_1, .. Q_n) although there is no necessity to express them in Darboux coordinates) the Hamiltonian flow is a vectorfield (i.e. a first order differential operator) defined by

df  = i_{X_f}\omega i.e. 

df(Y) = Y(f) = (1/2) \omega(X_f, Y) - \omega(Y,X_f) = \omega(X_f, Y) 

in particular in Darboux coordinates

X_f = \frac{\partial f}{\partial X} \frac{\partial}{\partial P} - 

          \frac{\partial f}{\partial P} \frac{\partial}{\partial X}  

This is usually used for the Hamiltonian H. In Darboux coordinates the flow of H is then 

\dot X = \frac{ \partial H}{'partial P},   \dot P = - \frac{\partial H}{\partial X}

which are traditionally called the Hamilton equations.

The Hamilton vectorfield is used to define the Poisson bracket

{f, g} = X_f(g) = - X_g(f) = \omega(X_f, X_g)

The Poisson Bracket allows an alternative, perhaps more familiar way to define the Darboux coordinates: coordinates P_i and Q_j (i, j = 1, n} such that

{P_i, Q_j} = \delta_{ij}

this should remind you of Quantum mechanics and indeed the Dirac quantisation principle can be formulated as saying that the first order part of  the quantum mechanical commutation relation as hbar ->0 is the Poisson bracket (although to make this precise turns out to be devilishly difficult)

 The symplectic form can be used to define a canonical volume form on phase space the Liouville volume form \lambda = \omega^n with the dimension of action. It therefore makes sense to say that a a subset of phase space has volume (i.e. action) 1 as a dimensionless number independent of the choice of coordinates and a fortiori of units of measurement. This is the geometric bases for the existence of a natural quantum of action (i.e hbar). The new SI system is redefining the kg by defining hbar := 1.0545718 × 10-34 m^2 kg / s thereby defining the kg (the metre is already defined as 3.3..... light nano seconds, so SI will effectively just define the (rather high) frequency f of a photon with an energy hf = 1kg c^2 )

The symplectic form \omega and a fortiori  the Liouville measure \lambda is invariant under the Hamiltonian flow since

\Lie_{X_f} \omega = (i_{X_f} d + d i_{X_f}) \omega

                             = d i_{X_f} \omega = ddf = 0

\Lie_{X_f} \lambda = \Lie_{X_f} \omega^n

                              = n\omega^{n-1} \Lie_{X_f}\omega = 0

It follows that \omega and \lambda are invariant under the flow of the Hamiltonian used to describe  the dynamics of a physical system. However, it also shows that \omega and \lambda are invariant under (infintessimal) rotations and translations on ordinary space if (as is typically the case) the infinitessimal  rotations or translations can be written as Hamiltonian vector fields when lifted to phase space (in this case, the cotangent bundle of Euclidean 3 space). E.g. the vector field on phase space defined by infinitessimal rotations along the z-axis can be written as Poisson brackets with the function on phase space (called the generator)

J_z (p,q) = xp_y - yp_x

which is the z-component of the angular momentum