Any shared hosting account or vps/dedicated server will do as long as it supports php, asp or .NET(aspx).
Build 496 of HDFVR introduced h.264 video recording which means that your users will need Flash Player 11.1 or above to run HDFVR.
According to statistics released by Adobe Flash Player is already installed on 97-99% of the computers connected to the Internet.
Just as web servers (Apache is a web server) serve static HTML pages and images, media servers (like FMS, Red5 and Wowza) can record and deliver video/audio content to Flash Player applications (like HDFVR).
The first media server was done by Adobe and it was called Flash Communication Server. It is now called Flash Media Interactive Server and it costs around $4500 to buy in the US (we know! It's a lot!). It has good documentation and there are a lot of articles and books explaining all things FMIS.
Red5 is an open source and free alternative to FMIS developed in Java.
Wowza is a commercial alternative (also developed in Java) to FMIS that costs about $995, comes with great documentation, a forum and good support.