Why are topological insulators interesting? Why are topological insulators interesting? Meaning, why should an undergraduate or graduate student start working on this? What are the technological applications? I am not sure how to answer these questions and wikipedia does not help since it does not explain why so many people work on this. I am especially interested in applications through photonics but any answer would be appreciated. Thanks in advance.
 A: Well, why things are interesting is a very subjective matter. What I can offer is two possible non-exclusive answers: from a theoretical point of view and one interested in applications:
Theoretical
Topological insulators are the most dominant example of phase transitions based on topology and not symmetry-breaking. Take the most famous topological insulator there is - the quantum Hall bar. Its insulating behavior is based on its topology, and small local perturbations cannot change it. This is unique and interesting, as it give rise to a lot of new neat theory: the field-theory that describes the insulators is topological, and these invariants can be read off it. The boundary between a topological insulator and a trivial insulator necessarily has an edge mode, because this boundary must have a gap closure. This edge mode is very robust, and unique in its properties. For example, in the quantum Hall effect, it is a 1d mode that cannot be written on a 1d lattice. This mode has something that is called a quantum anomaly, a form of symmetry breaking that reflects that fact that it lives on the boundary of a higher dimensional theory. All these things are very neat, and when you throw interactions inside you can get a lot of really interesting and challenging physics, and basically it is a huge playground with many tools.
Applicative
The edge modes of the topological insulators are interesting in that they are robust, and they are protected by a topological gap $\Delta$. As they are robust against local disorder and low-energy excitations, they often have very long life-times, $\tau \sim \exp(-\Delta/T)$. On top of that, they often manifest low-energy degeneracy, which makes them non-abelian anions. Therefore, they offer a promising building-blocks for quantum computation. They can (potentially) maintain coherence over long times, and can sometimes be used directly to build quantum logical gates, due to their non-trivial braiding properties. If I'm not mistaken Microsoft is investing quite a lot into this direction of building a quantum computer.
A: Almost as interesting as sharks with laser beams on their heads are topological lasers.  Topological protection can make microreasonators work in unison [1].
[1] Bandres, Miguel A., et al. "Topological insulator laser: Experiments." Science 359.6381 (2018): eaar4005.
A: This is to complement yu-v answer.
As this is not an "easy" topic (I mean, the effects are not quite seductive at simple sight compared to the sexy name and invariants are always presented in research papers in, to my taste, an obscure manner), a good reference is an appropriate starting point. 
One suggestion to start is Tkachov. It is very basic and you only need the courses of Quantum Mechanics; perhaps at graduated level to have a better grasp of the contents. 
Asbóth is also a good one, but it appears that contains some typos in the derivations.
The fun could start with Prodan's book, but it's aimed for a mathematical-mature audience. 
I would even dare to recommend Kitakawa review. It is aimed for topological effects in quantum walks, but the rudiments of this latter are quite straightforward.
Perhaps it is wise to survey first at the literature and then come back and ask a more precise question on the topic.
