In theoretical physics, the Dirac monopole is another name for the magnetic monopole; the article about the magnetic monopoles contains a lot of useful information.
In 1931, Paul Dirac realized that there is a symmetry between the electric fields and the magnetic fields in Maxwell's equations. However, only the electric charges appear as charged monopoles; the magnetic sources only appear as magnetic dipoles: the Southern pole of a magnet cannot be separated from the Northern pole.
Dirac investigated the question whether the magnetic sources can appear as monopoles. He considered a point-like magnetic charge whose magnetic field behaves as μ / r2 and is directed in the radial direction. Because the divergence of B is equal to zero almost everywhere, except for the locus of the magnetic monopole at r = 0, one can locally define the vector potential such that the curl of the vector potential A equals the magnetic field B.
However, the vector potential cannot be defined globally precisely because the divergence of the magnetic field is proportional to the delta function at the origin. We must define one set of functions for the vector potential on the Northern hemisphere, and another set of functions for the Southern hemispheres. These two vector potentials are matched at the equator, and they differ by a gauge transformation. The wave function of an electrically charged particle (a probe) that orbits the equator generally changes by a phase, much like in the Aharonov-Bohm effect. This phase is proportional to the electric charge qe of the probe, as well as to the magnetic charge qm of the source. Dirac was originally considering an electron whose wave function is described by the Dirac equation.
Because the electron returns to the same point after the full trip around the equator, the phase exp(iφ) of its wave function must be unchanged, which implies that the phase φ added to the wave function must be a multiple of 2π:
This is known as the Dirac quantization condition. In certain units, the condition is exactly given by the relation above. The hypothetical existence of a magnetic monopole would imply that the electric charge must be quantized in certain units; also, the existence of the electric charges implies that the magnetic charges of the hypothetical magnetic monopoles, if they exist, must be quantized in units inverse to the elementary electric
charge.
If we maximally extend the definition of the vector potential for the Southern hemisphere, it will be defined everywhere except for a semi-infinite line stretched from the origin in the direction towards the Northern pole. This semi-infinite line is called the Dirac string and its effect on the wave function is analogous to the effect of the solenoid in the Aharonov-Bohm effect. The quantization condition comes from the requirement that the phases around the Dirac string are trivial, which means that the Dirac string must be unphysical. The Dirac string is merely an artifact of the coordinate chart used and should not be taken seriously.
The Dirac monopole is a singular solution of Maxwell's equation (because it requires removing the worldline from spacetime); in more complicated theories, it is superseded by a smooth solution such as the 't Hooft-Polyakov monopole.
Mathematical details
Classically, gauge theory is described by a connection over a principal G-bundle over spacetime. Ordinary spacetime has the topology of R4, which is topologically trivial. So, the space of all possible connections over the principal G-bundle is connected. But let's see what happens when we remove a timelike worldline from spacetime. The resulting spacetime is homotopically equivalent to the topological sphere S2. So, it suffices to classify the connected components of the space of all connections over a principal G-bundle over S2. To do this, consider covering S2 by two charts, each homeomorphic to the open 2-ball such that their intersection is homeomorphic to the strip S1×I. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle S1. So, a topological classification of the possible connections is reduced to classifying the transition functions, which is given by the first homotopy group of G. In other words, a gauge theory can only admit Dirac monopoles provided G isn't simply connected. for instance, U(1), which has quantized charges isn't simply connected and can have Dirac monopoles while R, its universal covering group, is simply connected, doesn't have quantized charges and does not admit Dirac monopoles even in principal.
It's easy to see how this argument generalizes to d+1 dimensions with 
. We look at the homotopy group πd-2(G).