I know it a network attached storage and have worked with them in IT. My doubt is that if I have a NAS which is connected to my LAN port of internet router, then how can I access the data on it through Internet.
you access, NAS, via IP. If both your PC and NAS are connected to same lan, you just go to , run and type \\192.168.x.y (ip of you NAS) and it will open the NAS explorer with the shares you have created.
Maybe offtopic,but thats like exposing your entire network connected devices to the internet, read hackers.Doable by tweaking the firewall esp with samba but risky and not recommended.
If you have Synology NAS, then using the Synology QuickConnect URL you can access your NAS from anywhere. However you will have to use the Syno apps to access your data. It works well for Music and photos.
If your internet router has file sharing option, then you can create share via the internet router GUI and publish it. Thats the safe option and it will be available via wire and wireless. also you can secure it with router firewall. Lets know the the internet router make/model details to help further. I use the NAS share via wireless router this way.
Samba does not work well over internet (even with vpn)
I use owncloud for this purpose. It has a very friendly interface for file management. It is not very hard to setup and it is not very resource hungry. Not very familiar with your platform so not sure whether binary packages are available.
If you go with owncloud simply do a port forward in your internet router from a high port (to minimize security issues) to whatever port you are running owncloud internally