If you are interested in tangles in the sense of Robertson and Seymour, this is just to provide some perspective on it. I am working on this for my Ph.D. project and I thought maybe it is considered helpful if I share this high-level, intuitive perspective here (it is not a detailed definition):
The best and shortest description, I think, is given in the following quote. It mentions the proof of the graph minor theorem, but in fact it is the same concept of tangles as in the proof of the graph structure theorem:
Quote: „Originally devised by Robertson and Seymour as a technical device for their proof of the graph minor theorem, tangles have turned out to be much more fundamental than this: they define a new paradigm for identifying highly connected parts in a graph.
Unlike earlier attempts at defining such substructures—in terms of, say, highly connected subgraphs, minors, or topological minors—tangles do not attempt to pin down this substructure in terms of vertices, edges, or connecting paths, but seek to capture it indirectly by orienting all the low-order separations of the graph towards it.
In short, we no longer ask what exactly the highly connected region is, but only where it is. For many applications, this is exactly what matters.
Moreover, this more abstract notion of high local connectivity can easily be transported to contexts outside graph theory. This, in turn, makes graph minor theory applicable beyond graph theory itself in a new way, via tangles.“ End of Quote.
This quote is from Reinhard Diestel, in the preface to the 5th edition of his Graph Theory book.