I have both rooted and a non-rooted phones, there are some apps in my rooted phone which I took a backup using 'Titanium Backup' can I restore it on my non-rooted phone using any method?
It depends on what kind of backup you want to restore. If it's an XML backup (bookmarks, messages, call log or Wi-Fi access points) then you can restore it. In the case of app+data pairs, only the app can be restored (without its data), and that is only if it's a non-system app.
XML backups can be restored by pressing the Menu button, then scrolling down and selecting the option 'Restore data from XML'.
Non-system apps can be restored by choosing their respective entries in the 'Backup/Restore' tab, selecting the desired backup, then picking the 'App only' button from the popup.