What are skyrmions?

Skyrmions are described somewhat impenetrably here on wikipedia, and they are rather difficult beasts to summarize briefly, but I'll give it a go.  There are a number of physical systems with internal degrees of freedom that can be described mathematically by some kind of vector field.  For example, in a magnetically ordered system, this could be the local magnetization, and we can imagine assigning a little vector \( \mathbf{m}\) (that you can think of as a little arrow that points in some direction) at every point in the system.  The local orientation of the vector \( \mathbf{m} \) depends on position \(\mathbf{r}\), and there is some energy cost for having the orientation of \( \mathbf{m} \) vary between neighboring locations.   In this scenario, the lowest energy situation would be to have the direction of \(\mathbf{m}\) be uniform in space.

Now, there are some configurations of \( \mathbf{m}(\mathbf{r})\) that would be energetically extremely expensive, such as having \( \mathbf{m}\) at one point be oppositely directed to that at the neighboring sites.  Relatively low energy configurations can be found by spreading out the changes in \(\mathbf{m}\) so that they are gradual with position.  Some of these are topologically equivalent to each other, but some configurations of \(\mathbf{m}\) are really topologically distinct, like a vortex pattern.  Examples of these topological excitations are shown here.   With a lone vortex, you can't trivially deform the local orientations to get rid of the vortex.  However, if you combine a vortex with an antivortex, it is possible to annihilate both.

Skyrmions (in ferromagnetis) are one kind of topological excitation of a system like this.  They are topologically nontrivial "spin textures", and in real magnetic systems they can be detected through techniques such as magnetic resonance.   It's worth noting that there are other topological defects that are possible (domain walls that can be soliton-like; defects called "cosmic strings" if they are defects in the structure of spacetime, or "line defects" if they are in nematic liquid crystals; monopoles (all arrows pointing outward from a central point) sometimes called "hedgehogs"; and other textures like the boojum (a monopole pinned to the surface of a system; relevant in liquid crystals and in superfluid ³He)).  With regard to the last of these, I highly recommend reading this article, which further cemented David Mermin as a physics and science communication idol of mine.