Barzakh is the world after this world which starts from death and goes on till the day of Judgment. It is mentioned in Quran (There is a Barzakh after their death till the day of they will be resurrected) (Sura 23, Verse 100).
Barzakh includes the grave but it is wider than the grave. Grave starts on burial but Barzakh starts on death which is before burial.
The pious believer's grave is a garden from the Paradise while the grave of the disbeliever is a ditch from the hellfire, as the Hadeeth states.