Well, depends on how you define alias.
You said this:
"Thanks but that's not what I wanted. What I really want is that a user types for example alias1.com and it will redirect them to domain.com. How can I make this happen?"
If that's the case, then you have to register alias1.com as another domain.
If you mean something like:
alias1.domain.com
then no, you don't have to register another domain because alias1 is a subdomain of the domain.com domain.
Anything at the top level (actually one level down, as the Top Level Domains are .com, .org, .net, etc) (ie foo.com, bar.com, test.com, blah.net, glad.org, etc) will have to be registered as a domain through a valid registrar.