Also, how does it affect our websites? Do we need to shift all IPv4 websites to IPv6 or both?
There should be no need to shift existing websites onto IPv6 for a long time. IPv4 will keep running for years yet. There will be a process where new services begin to use IPv6 instead of IPv4. The two systems will co-exist side by side. To the end user, assuming they have their software up to date, they should notice no difference. If you want to test whither your PC can cope with IPv6, just go to this page: ipv6test.google.com
As IPv4 addresses run out, there will not be a sudden end. Rather the price to obtain an IPv4 will increase and IPv6 will begin to look attractive as an alternative.
8 June, 2011 was IPv6 day. Many major sites including Google, Facebook, Yahoo! and others turned their servers to IPv6 for one day. The internet continued as normal that day.
If you are running your own web server, you will have to read up some details on IPv6 in order to use it, when the time comes. There are a few tweaks that will need to be made by the admin to the server. However, if you, like most web site owners, purchase your hosting from a web hosting provider, then you needn't do anything. The service provider will take care of everything.
Why is IPv6 not backward compatible? Well IPv6 is more than just a change in the IP number though that is a significant part of it. There are a load of other changes to the function of the protocol which should enhance security and speed while simplifying many operations. Have a look at this article on IPv4-IPv6 Compatibility
IPv6 is mostly about the "plumbing" between your server and the user. The various parts of that plumbing may or may not be IPv6 compatible. Even if you make sure the network from your server to your ISP is your ISP may not support it. On the other hand, it's worth making sure your webserver software (Apache or IIS say) supports it (Apache v2.0 and upwards does, IIS 6.0 and upwards does), and your network equipment.
The main way that it would stop things working is when someone registers a domain and there are no IPv4 IP addresses available for the hosting company to use, so they only get an IPv6 IP address. Then it will only be accessible to people who have IPv6 all the way between them and the server.
Although the last lot of IPv4 addresses have been given to the people who give them out, they haven't all been given out. Also, if it really looks like it's getting short people may release unused addresses.