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.