Loosely speaking, it's a structure that can't be continuously transformed away. For example, consider a (frictionless, non-dissipative etc.) fluid flowing in a circle around a given point: a vortex. Since this has net angular momentum, and a fluid moving uniformly in one direction doesn't, it's impossible for the vortex to dissipate. We can move it around, or split it up, or fuse it with other vortices, but we can't just smooth it out.

I think, most close point, to think of it like additional dimension.

For example, imagine we live on 2D infinite plane, and once found out, that in some places, moving on for example for 1km, we see same shift as with moving linearly (to be more exact, by some arc or some polygonal line) by other route for 100km (really need much less to prove topological defect, just standard scientific 3-sigma difference from our model is enough), so this mean, we have crease on space-time topology (and possible, some sort of wormhole, or just holes on plane and controllable way to navigate/move from one hole to other and move between hole and our plane).