The following steps will help you to migrate your Joomla 1.5. website to Joomla 2.5
Make sure that your Joomla 1.5 is upgraded to the latest stable version of Joomla (1.5.26)
Backup your Joomla 1.5 website using a free component Akeeba Backup from https://www.akeebabackup.com/ or you may also manually backup your Joomla website by going to phpmyadmin and export the dump of your mysql database and then create a backup of Joomla files by going to Cpanel > File Manager, select all files/folders and click compress to create a zip file.
Download JUpgrade Component by going to Joomla Extensions Directory or by following this direct linkhere. This component requires registration to download. (Latest Version Available is 2.5.2).
Go to your Joomla 1.5 Administrator and go to Extension >Install/Uninstall and then install JUpgrade component.
Go to Components > JUpgrade and click Start Upgrade Button.
jUpgrade component will download the latest version of Joomla 2.5, decompress it and install it. After that, it will migrate all of your data from old Joomla 1.5 to the new Joomla 2.5 installation.
jUpgrade will install the new website in a sub folder jupgrade e.g yourdomain.com/jupgrade
The new site should have all the content, users and other core components data.
Move the existing Joomla 1.5 website from root to another subfolder(e(.g /oldsite).
Finally move your Joomla 2.5 website from jUpgrade folder to root to make it live.