The answer depends on what package you have available in terms of features, storage space and bandwidth with your current host. Often there is sufficient space available to add another domain using your existing hosting package, usually called an 'add on' domain. Then if your traffic isn't too high you will also usually have sufficient bandwidth.
You need to check with your current host that you can add a further domain and that there is enough space and bandwidth to cope with the demands of a second site. If there is it shouldn't cost you any more.
In terms of the set up, once you have created your 'add on' domain you will need to treat it like a brand new site (it is really) and install Wordpress on that domain as well. Once you have Wordpress installed then you just select a theme and continue setting up the site in much the same way as you did your original one. They are 2 separate sites.
One further point is that you will also need to register your second domain in its own right with a registrar, assuming you want a new domain and not a sub-domain of the original site. If it is a completely different topic it really should be a new add on domain rather rather than a sub-domain.
The sub-domain option can possibly be used instead of an add on domain, but it would have virtually the same domain name as your existing site, doesn't sound like that is what you want to do.
For example it might look like 'www.newdomain.myexistingdomain.com' whereas an add on domain would look like www.newdomain.com Much better for a different topic.
Hope that makes sense and helps explain.