You can host a forum from a home PC, but unless you have a pretty good connection I wouldn't bother. You'll need to install Apache, PHP, and MySQL on your box. There are several WAMPS (Windows Apache MySQL PHP) to do this such as AppServ or XAMPP (XAMPP also includes Perl).
PHP is a scripting language that pretty much all forum software uses. It needs an interpreter to run which is what "PHP 4.3.3" is, it's the version of the interpreter. MySQL is a database software using the SQL (structured query language) language. It is used to store users, posts, and many many other things in forums. 4.0.16 is the version of MySQL.
Keep in mind that both those versions you listed are quite out dated, and you should move on to PHP 5.2.6 and MySQL 5.0.67 or newer.
A host is quite cheap. You can get a host at GoDaddy.com for around 5$ a month I think that will do everything that you want. I personally use HostGator.com and I pay $9.99 a month, but I like to have the extra junk. Plus I thought GoDaddy was pretty slow, but it does have good value.
Also keep in mind that vBulletin costs like $180 a year, whereas phpBB is free.