What exactly is a pointer domain and how do you get one?

It's my understanding that it's a domain name that redirects to another site. Such as www.abcsite.com redirects to www.abcd.com, but anyone who types in either address will get to abcd.com - is that right?

Also, what is the difference between purchasing a domain name and registering one?

Thank you!

A "pointer" domain is correctly known as an alias

It's my understanding that it's a domain name that redirects to another site. Such as www.abcsite.com redirects to www.abcd.com, but anyone who types in either address will get to abcd.com - is that right?

Sort of, they will get the same content (a mirror site) at both URLs but unless one is redirected to the other they will be seen as independent entities.

Also, what is the difference between purchasing a domain name and registering one

Nothing!

and how do you get one

Register one and have the DNS point to the same server and the hostheader or VirtualHost pointed to the same physical HD space as the main site.

That is 301 redirect (on the server side). More used for SEO needs. I think that you shouldn't use only for have a fun.