FlowPlayer is self hosted flash player for websites. It supports direct content and streaming servers. Remember though, video requires a lot of bandwidth so make sure your servers/webhost can cope with that requirement.
If not, you might want to look at offsite hosting of the video content. YouTube is the obvious free one (and supports embedding the video on your site), but there are loads of others around if you Google for them.
For video editing, I'm a big fan of cinelerra, but it's Linux only. Windows movie maker does the basics if you just need something simple. Or there's virtual dub, but it only does clipping and doesn't do transitions or non-linear mixing last time I used it.
For processing the videos, ffmpeg and MEncoder (comes with MPlayer) both do all kinds of conversion,rescaling,encoding etc and will come in handy if you need to convert to specific formats or need to compress the clips a bit.