Addon domains are additional domains which are redirected to a subfolder in your main website. Luckily, even though your new website is located in a subfolder under your main website, it is possible to configure the addon domains so that your website visitors and search engines see these as independent websites.
First you need to register the domain from a domain registrar as usual. Then, if your webhosting provider allows this, you can setup this addon domain to point to a subfolder in your main account. This is usually done from the hosting provider's control panel.
When setup properly, your website visitors will never see the "real" domain name anywhere.
[Here is an example. Both this website and my other site www.easydonations.net are located at the same hosting account. Try and see if you can figure out which is the addon domain...]
But here comes the tricky part... Often times the addon domains functionality offered by webhosting providers doesn't work like this. Even though they call them addon domains, these domains actually behave like simple forwarded domains - with or without masking. This can make your websites look quite unprofessional (see problems with domain forwarding for details). Therefore always make sure you know what your hosting provider means by multiple or unlimited domains and try it out before committing yourself.
Here is a list of website hosting providers I have found through trial and error and know their addon domains function properly: website hosting providers. (I'm sure there are many more out there, but unfortunately I haven't found a way to tell them apart without actually testing them out.)
Which brings us to the last type of domain forwarding: dedicated IP addresses.