I recently read an article in which they wrote that that there will be a time when there will be no more IP addresses available. Is that right? Do I need to register a website now just in case?
No, you don't need to worry.
Registering a website (domain name) is totally separate to having an IP address. Registering a domain name just gives you the name, this name will then be entered into DNS servers, which will allow the name to be resolved into an IP address. This will be the IP address of the web server that holds your site. The company running the web server or their ISP will be responsible for ensuring that they have valid IP address, you don't need to worry about it. It can also change, so you don't need to reserve a specific address.
Something called Network address translation (NAT) has relived a lot of the pressure on IPv4 addresses. It allows a single external IP address to be mapped to a large number of IP addresses internally within a smaller network. (Such as a home or company network).
We are moving towards support for IPv6, which will expand the number of available addresses significantly. Most modern operating systems now support it, and a a lot of network hardware is starting to support IPv6 now.
Unless you are a registrar or ISP you do not need to significantly worry about IpV4 addresses running out. Just consider it when buying hardware and try to buy IPv6 compatible hardware where possible.
You can read more about it here.
Probably not as NAT has freed up a lot of IP addresses, plus once they are scarce the price they command will rise to the point that people will start selling them.
It's the same reason that all commodities in a market are available at some price. Once the price is high enough it tempts people to sell them, find ways to extend them (NAT, Apache Virtual Hosting etc)
I'm assuming that IP addresses can be bought and sold. Not sure if they actually are at the moment.
Also having a web site (you probably meant domain name) doesn't automatically get you an ip address. Your hosting provider will probably use one IP address to host many domains at once, and rely on the header your browser sends to determine the domain name requested.
Well, people are crying since some years that all IPv4 addresses are used up...as I said, they're saying it now for years. The truth is that 90% of all IPv4 addresses are allocated, but the rest will hold for another 2 to 3 years...and after that there's IPv6. Maybe you wanna read the slashdot story about it.