I'm pleased to hear that you like CiviCRM. It is a good piece of software that has been used by many non-profits across the world.
However, I think you need to be realistic about the cost of installing a CRM system like this. While CiviCRM is 'free' in the sense that it is open source and you can download the code and alter it as you see fit there are still costs associated with using it.
CiviCRM is a complex piece of software and unless you have the technical skills yourself you will likely need to employ a professional to set up the software for you. You will also need to pay for ongoing hosting costs for the software. I would also recommend that you consider training costs for yourself and your staff/volunteers that will be using it.
I'm afraid that no CRM system is entirely free of cost.
I would suggest that you start by looking at the Getting Started section of the CiviCRM website.
In answer to your specific questions about hosting. Yes, it would need to be hosted on a web server (it is possible to host it on your own computer but this is unlikely to be a useful option in practice). Web hosting means purchasing space on a server (computer) that you can use to host a website. You may be able to host it with your current website, but CiviCRM requires significantly more resources than a standard website so you may need to upgrade your hosting package to accommodate this. Hosting CiviCRM is quite a technical matter so I would recommend that you get advice about this.
If you host CiviCRM with your website it would share the same domain name. If you host it separately it would need its own domain name (or a sub-domain of your domain).
Alternatively there are providers who offer CiviCRM hosting packages and would be able to take care of all these technicalities for you - see Hosting Providers.