A higher site's coverage satisfies
If $\left\{f_: U' \rightarrow U\right\}$ is a covering family and $g: V \rightarrow U$ is a morphism, then there exists a covering family $\left\{h: V' \rightarrow V\right\}$ such that the composite $g\circ h$ factors through $f$ up to higher morphisms.
instead of the usual condition where strict equality $g\circ h=f$ is involved.
Also a higher sheaf takes value in higher categories, for example, $(2,1)$-sheaf is a stack (category fibered in groupoid).
I understand the higher descent when the site is a regular one (the $3$-cocycle codition involved in the descent for usual stack is nothing but a $2$-morphism and when the sheaf goes higher the $n$-cocycle codition gets imposed; see http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/descent#AsGluing).
But how exactly is a higher sheaf on a higher site different from that on a regular site?