There are two methods of copying your webhosting account from one Server to another. You probably want to do it because you need to transfer your website on different Server.
First tutorial is for Shared Hosting (When you have only Shared Hosting Account on both the servers)
1. Loin to your cPanel of the site you want to copy.
2. Goto : Backups
3. Download Partial Backups (Home Directory, Databases, E-Mail Filters, E-mail forwarders), you can take full backup but unless you have root access on second server its useless.
4. Now login to the cPanel of second shared account.
5. Goto Restore and Restore all the files you downloaded earlier.
It will take some time so don't hurry.
6. Then Change the nameservers of your domain name.
If you have root access on the second server, then you can try the simple method which is more effective.
1. Login to WHM on the second server using your root account
2. Goto : Transfers->Copy an Account from another server using account password
3. Now give the following details :
Server to copy from (ip or FQDN):
Domain Name* :
Username to copy: :
User's password: :
Give new account an ip address :
Permit fallback to old system (see above):
4. Then click on copy account
5. WHM will automatically copy the account from your old account and all you will need is to change the nameservers of your domain name.
The second method which requires root access is the best one as it transfer all the account files automatically and you will not need to edit your database files or anything. But if you transfer the account using First method then you might need to edit the configuration files of your database if your account username is different on previous site.
Some host provide you service of restoring/transferring your website from your old account, you can ask your new host to do it for you. He will happily do it (he can't do only if he don't have root access, that means he is also a Master Reseller/Reseller of some other hosting site)
If you face any problem then you can ask here.