Regarding Local Hard disks
 

In Fedora when we double click the Partitioned Local Hard Disks then we have to give the root password otherwise it will not open. Is there any way to read and execute the Partitioned Local Hard Disks without giving the Root Password.

Why would you want to? Security models and conventions exist for very good reasons, and I'd really suggest trying to react against them. Why would you ever want to provide unsecured access to the partitions on the system? It's not even as if you would be automating this or going in there with any regularity at all. It's possible to do it via sudoers or something, but I'd urge you to decide you don't want to after all.

rdtech25, acid_kewpie is correct, you don't want to give unsecured access to the ability to modify or alter any partition information. However, I read your question a little differently and I am wondering if you are trying to MOUNT an existing partition and that this is requiring root or sudo privilege. If this is the case, you can add the partition entry to your /etc/fstab which will allow mounting of this partition without elevated privilege.

Oh yes, if I read that again it does sound like it might refer to existing disks. Don't recognise the text in the slightest though.... where is this coming from?

Thanks to all of you for replies. Now I will add the partition entry to my /etc/fstab which will allow mounting of this partition without requiring any kind of privilege...but sorry to say that I don't no how to do it.

The file is located in /etc/fstab. You will pretty much add an entry to it containing the same data as your manual mount statement. For example, to mount sda2 on /mnt/mydrive you would have an entry similar to this:
I would recommend that you read the man pages (man fstab) for clarification on the meaning of the parameters for your mount as what I showed above is just an example.