If you want your own domain (e.g. www.bawn.ie) then you'll have to buy it in the first place and pay a hosting company for the privilege of some web space.
Under this model you'd probably create a web site from scratch, using a protocol called FTP to get your web pages from your PC to the web space. The web pages could be crafted in a text editor (a bit technically involved) or created using a software package such as Microsoft FrontPage or Macromedia Dreamweaver.
Having said that, one of the host companies I've used quite a bit has some sort of built in tool so that you can create your web pages on-line. I've never used it, but I guess it might be what you're looking for. This is them:
This company is also OK:
There are also some web sites that are specifically for genealogists to post their family trees on, but you probably know about that sort of thing already.
K x