Example: Irreducible component - affine varieties 
Again, I know how to prove the statement. But, I cannot find any example. Please help me for finding an example. Thank you:) 

If $n=2$, take for $X$ the circle $x^2+y^2=1$, for $H$ the "hypersurface" $y=0$ ( a good old line!) and then  $X\cap H$ consists of the two irreducible  components (=points)  $\{(-1,0)\}$ and $\{(1,0)\}$.
