I find cPanel with one of the hosts has PostGRE databases while other hosts don't have it. What is PostGRE database and what is its significance (if any) ?
Where can it be used?
PostgreSQL is another relational database like mySQL. It's a bit speedier in some areas, and slower in others.
The reason you don't see it in many hosts these days, is because cPanel 10's support was pretty poor--it was there, but there were some stability issues that I heard a lot about from some other hosts.
It is an alternative, if the program you're using supports it. For example, in PHP, when you send the commands to mySQL, you would send slightly different commands to postgreSQL because of the syntax differences.
But, make no mistake about it, mySQL is much quicker for the type of web applications that make a lot of database queries. It can never hurt to learn.