One of the peculiar (and somewhat appealing) features of quasi-categories is that many properties from ordinary category theory characterized equality are characterized by some form of homotopy (morphisms to a terminal object are homotopy equivalent instead of unique, associativity is up to homotopy, and so forth).
While I know what homotopy is in the sense given by any introductory algebraic topology textbook, and can generalize this to "some kind of congruence relation on morphisms" in a 1-category, I'm not clear on what homotopy of $n$-cells in a quasicategory should be and I can't seem to find a definition. Any pointers?
If I understand the question correctly, you should take a look at the definitions of the right and left mapping spaces between two objects in a quasi-category. The 0-simplices in this simplicial set are the morphisms, 1-simplices are the homotopies between morphisms (in the sense of Boardman and Vogt, see their construction of the homotopy category), and so on. As Zhen Lin said, this should be covered in any introduction to quasi-categories. Aside from Lurie I would highly recommend the course notes of Denis-Charles Cisinski, available on his web page.