If I signed up with Bravehost to host my website, should I be able to build the website there? Or do I build it on my pc with some tool (what?) and then upload it?
It is possible to build your website online, if this is just a shortterm temporary thing. (Example, something you link to in your emailed party invitation).
Doing it on your computer is easier and better. There are hundreds of programs you can use, including Notetab, Editpad, Arachnophilia and HTML-kit.