I understand 192.168.x.x is our local IP address, and the public IP is some other. I used this site to get my IP, and the result: 
But in my official ISP's site and my Router's login, it is as follows:
My router and ISP shows my an IP, while the website shows me another IP. What is my actual IP address? And why am I assigned multiple IP addresses?
I am asking so that another computer can establish a connection with my computer, as I wrote a java program for that
Your internet provider is using Carrier grade NAT. Your local (to your house) IP address is 192.168.x.x. Your local (to your ISP) address is 10.230.x.x. The public, routable IP which you are sharing with many other customers of your ISP is 49.207.x.x.
You are assigned a local IP for your internal network, and a public IP for the internet, if you type into google "what is my ip" it will return your public IP address and when you go to cmd and type "ipconfig" it will show a IPv4 address, that is your internal/local IP.
Local IPs tend to be something like:
10.x.x.x,
172.16.x.x
192.168.x.x
One of the reasons you need a local IP is due to something called NAT or Network address translation.
Further Reading: https://en.wikipedia.org/wiki/Private_network
You can connect two computers in your house to each other via the 192.168.x.x IPs.
To have the outside world connect to your PC is something your ISP / Cable company is trying to prevent; for instance generally the ISP will not forward port 25 for incoming mail.
They will sell you a fixed IP that you can use to have Internet access TO your PC, but you will need a server security hardened to handle everything that comes in, including various attacks that happen when you connect to the Internet, as you will be a live site, accessible to anyone. This means firewalls need to be set up correctly, etc.
So, for the average cable user, the answer is "you can't".
