Topological Charge. What is it Physically? I have seen the term topological charge defined in an abstract mathematical way as a essentially a labeling scheme for particles which follows certain rules. However I am left guessing when trying to explain what physical properties of a system lead to the need to introduce this new type of "charge." 
If it makes any difference, I'm interested in studying the statistical properties of Quantum Hall systems (and in particular anyonic interferometry), where the different topological charges contribute to the total number of quantum states. 
 A: Local quasiparticle excitations and topological quasiparticle
excitations
To understand and classify anyonic quasiparticles in topologically ordered states, such as FQH states, it is important to
understand the notions of local quasiparticle excitations and topological
quasiparticle excitations.  First let us define the notion of ``particle-like''
excitations.  
Let us consider a system with translation symmetry.  The ground
state has a uniform energy density.  If we have a state with an excitation, we
can observe the energy distribution of the state over the space.  If for some
local area the energy density is higher than ground state, while for the rest
area the energy density is the same as ground state, one may say there is a
``particle-like'' excitation, or a quasiparticle, in this area.  Quasiparticles
defined like this can be further divided into two types.  The first type can be
created or annihilated by local operators, such as a spin flip.  Hence they are
not robust under perturbations.  The second type are robust states. The higher
local energy density cannot be created or removed by any local operators
in that area.  We will refer the first type of quasiparticles as local
quasiparticles, and the second type of quasiparticles as topological
quasiparticles.
As an simple example, consider the 1D Ising model with open boundary condition.
There are two ground states, spins all up or all down.
Simply flipping one spin of the ground state leads to the second excited state, and creates a local quasiparticle. 
On the other hand, the first excited state looks like a domain wall.
For example the spins on the left are all up while those on the right all down,
and the domain wall between the up domain and the down domain is a topological quasiparticle.
Flipping the spins next to the domain wall moves the quasiparticle but cannot remove it.
Such quasiparticles is protected by the boundary condition.
As long as as the two edge spins are opposite, there will be at least one domain wall, or one topological quasiparticle in the bulk.
Moreover a spin flip can be viewed as two domain walls.
From the notions of local quasiparticles and topological quasiparticles, we can
also introduce a notion topological quasiparticle types (ie topological charges), or simply,
quasiparticle types.  We say that local quasiparticles are of the trivial type,
while topological quasiparticles are of non-trivial types.  Also two
topological quasiparticles are of the same type if and only if they differ by
local quasiparticles.  In other words, we can turn one topological
quasiparticle into the other one by applying some local operators.  The total
number of the  topological quasiparticle types (including the trivial type) is
also a topological property. It turns out
that this  topological property is directly related to another topological
property for 2+1D topological states: The number of the  topological
quasiparticle types equal to the ground state degeneracy on torus. This is one
of many amazing and deep relations in topological order.
See also Why is fractional statistics and non-Abelian common for fractional charges?, A physical understanding of fractionalization, and Whis is the difference between charge fractionalization in 1D and 2D?
A: The distinction between "ordinary" and topological charges comes from the fact that the conservation of the ordinary charges is a consequence of the Noether's  theorem, i.e., when the system under consideration possesses a symmetry, then according to the Noether's theorem, the corresponding charge is conserved.
Topological charges, on the other hand, do not correspond to a symmetry of the given system model, and they stem from a procedure that can be called topological quantization. Please see the seminal work by Orlando Alvarez explaining some aspects of this subject. These topological charges correspond to topological invariants of manifolds related to the physical problem.
One of the most basic examples is the Dirac's quantization condition, which implies the quantization of the magnetic charge in units of the reciprocal of the electric charge. This condition is related to the quantization of the first Chern class of the quantum line bundle. It is also possible to obtain the quantization condition from single-valuedness requirement of the path integral. The existence of the topological invariants is related to a nontrivial topology of the manifold under consideration, for example nonvanishing homotopy groups, please, see the following review  by V.P. Nair.
Of course, topological charges can also be non-Abelian; a basic example of this phenomenon, is the 't Hooft-Polyakov monopole, where these solutions have non-Abelian charges corresponding to weight vectors of the dual of the unbroken gauge group. Please see the following review by Goddard and Olive.
It should be emphasized that the distinction between ordinary charges and topological charges is model dependent, and "ordinary" charges in some model of a system emerge as topological charges in another model of the same system. For example the electric charge of a particle can be obtained as a topological charge in a Kaluza-Klein description. Please see section 7.6 here in Marsden and Ratiu.
Topological charges correspond sometimes to integer parameters of the model, for example, Witten was able to obtain the quantization of the number of colors from the (semiclassical) topological quantization of the coefficient of the Wess-Zumino term of the Skyrme model.
A simple example, where quantum numbers can be obtained as topological charges is the isotropic harmonic oscillator. If we consider an isotropic Harmonic oscillator in two dimensions then its energy hypersurfaces are $3$-spheres, which can be viewed as circle bundles over a $2$-sphere by the Hopf fibration. The $2$-spheres are the reduced phase spaces of the (energy hypersurfaces) of the two dimensional oscillator. In the quantum theory, the areas of these spheres need to be quantized, in order to admit a quantum line bundle. This quantization condition is equivalent to the quantization of the energy of the harmonic oscillator.
Actually, these alternative representations of physical systems, such that ordinary charges emerge as topological charges offer possible explanations for the quantization of these charges in nature (the Kaluza-Klein model for the electric charge, for example).
A current direction of research along to these lines is to find topological "explanations" to fractional charges. One of the known examples of is the expanation of the fractional hypercharge of the quarks (in units of $\frac{1}{3}$), which can be explained from the requirement of the anomaly cancelation (which is topological) of the standard model, where the contribution of the quarks must be multiplied by $3$ (due to the three colors). In addition to anomalies, it is known that the existence of fields of different irreducible representations in the same model and separately knotted configurations may give rise to fractional charges.
