I have been studding about compatification of a topological space $X$. But I have low understanding about the Stone-Cech compatification, specially construction of the Stone-Cech compatification on natural numbers.(I mean $X = \beta\omega$). I have some difficulty with construction of $ \beta\omega$ and $\beta\omega - \omega$.

1) : Can you give me brief and to the point explain about $ \beta\omega$ and it`s members?

2) : What kind of ultrafilter does put on $\beta\omega$?

3) How are the members of $\beta\omega - \omega$? Which relationship is there between a ultrafilter and members of $\beta\omega - \omega$?

Thanks.