Look into ustream.com, simple to setup, and they have tutorials and hardware suggestions. YouTube, MikeTV and a few other sites also have live streaming options.
You are not really "recording" live events but broadcasting (streaming) them. Unless as the post above you are recording something that you are not authorized to record and re-broadcasting it, that is breaking laws.
Maybe I'm interpreting this incorrectly, but it sounds like you want to record live events that are being broadcast on television and then streaming it online for others to view--which would be illegal for most events, particularly commercial events like concerts, performances, and sporting events.