You should verify that you can connect using psql. Try psql -U username -h localhost dbname. It should prompt for a password then connect. Run SELECT postgis_version(); to verify that PostGIS is active in the database.
If you can connect but SELECT postgis_version() reports an error, PostGIS isn't installed in the database:
ERROR: function postgis_version() does not exist
LINE 1: SELECT postgis_version();
Other solution:
SELECT * FROM Postgis_lib_version();
If you want also to see the version of PostGIS and the version number of the libraries GEOS and Proj4 just change lib to Full.
If you get the above error, then presuming you're running PostGIS 2.0 on PostgreSQL 9.1 or above, connect as user postgres and run CREATE EXTENSION postgis;. Eg:
psql -U postgres -h localhost dbname -c 'CREATE EXTENSION postgis;'
You might also need to install some of the extras, like the PostGIS topology support:
CREATE EXTENSION postgis_topology;
or the legacy support script, which isn't packaged as an extension and must be sourced:
psql -U postgres -h localhost dbname -f /path/to/postgis-2.0/legacy.sql
See the PostGIS documentation.
If you can't connect to the DB, it's probably a pg_hba.conf issue, failure to create the DB or user, etc. Hard to say without error messages and log contents.
I am not sure if it matters, but for me, Postgis 1.5 and Postgres 9.0 (x86) are talking to QGIS 1.8.0 without problems.
QGIS is still 32-bit, even on Windows 7 64-bit machines. Give the x86 version a try, and the old postgis version.
Best Answer
It looks like you have confused the username and the database name.
If you created a database called shashi, you should have Database: shashi in the QGIS PostGIS Connection window. Your username is likely postgres.
If that doesn't work, look in the message log. My logs are in /usr/local/pgsql/var/, but they may be somewhere else on your machine.