Hello,
Laravel makes it very easy to manage your database connections through app/config/database.php.
As you noted, it is looking for a database called 'database'. The reason being that this is the default name in the database configuration file.
'mysql' => array(
    'driver'    => 'mysql',
    'host'      => 'localhost',
    'database'  => 'database', <------ Default name for database
    'username'  => 'root',
    'password'  => '',
    'charset'   => 'utf8',
    'collation' => 'utf8_unicode_ci',
    'prefix'    => '',
),
Change this to the name of the database that you would like to connect to like this:
'mysql' => array(
    'driver'    => 'mysql',
    'host'      => 'localhost',
    'database'  => 'my_awesome_data', <------ change name for database
    'username'  => 'root',            <------ remember credentials
    'password'  => '',
    'charset'   => 'utf8',
    'collation' => 'utf8_unicode_ci',
    'prefix'    => '',
),
Once you have this configured correctly you will easily be able to access your database!
Thank You!!