It began as blogging software and led/leads the field, and nowadays can do a lot more than just blogging. It describes itself as a "Blog Tool, Publishing Platform, and CMS"
A more powerful/flexible alternative, and is a classic solution for a medium-sized company or bigger, but I suspect that may be taking the pretence too far. On the one hand it's reputed to be complex to set up the way you want it. On the other hand it has, like Linux, spawned an ecology of pre-configured distros to suit many needs.
Both are very widely used and have thriving communities. I'd suggest you take a quick look at both before deciding whether WordPress is flexible enough or Drupal is straight forward enough.
Thinking about quality of service and availability, a home server might let you down at a crucial moment. Loads of hosting sites will stand you up a server for not much cash. I know that Amazon and Google are into cloud services, but I don't know how mature / free these are.