You'll need a webserver running PHP5.3+, preferably a Linux server. A shared host is not generally recommended (though
BlueHost has a really good business hosting plan if your site doesn't warrant anything larger; if you're looking at a VPS or Dedicated Server, I'd recommend my host
StableHost).
I have made a test script to ensure you won't have (many) issues in running a forum, this is available at:
UBB.threads v7.6.0+ System Requirement Test - 20151204.
Additionally you will need a domain name, there are many domain registries out there, including whatever webhost you end up at (in fact, some hosts include a free domain).