Boardman and Vogt invented the notion of weak Kan complex, Andre Joyal continued to show their relevance to category theory, and Jacob Lurie proceded to write a 600 page book as a foundation for the theory.
The overarching idea is that a simplicial set looks something like a category. The vertices look like objects, the 1-simplices look like morphisms, the 2-simplices look like commuting triangles.
But now think of each 2-simplex as "a witness to the commutativity" of the given triangle. In category theory, either a triangle either commutes or it does not, but in the theory of infinity-categories, the triangle may have multiple "ways" of commuting.
In this talk, I will discuss the basics of the theory of infinity categories. I will discuss the functors between infinity categories (which are just the maps of corresponding simplicial sets), the relationship between infinity categories and categories enriched over topological spaces, the homotopy category of an infinity category, the concept of homotopy commutativity vs. homotopy coherence, and finally give the definitions of limits and colimits of diagrams and give a nice example which brings it all together.