Flash still has a purpose when necessary, but most web developers would agree that the future of Internet video is HTML5. Two scripts I would highly recommend are Video.js and MediaElement.js. The first is much simpler but also limited with basic functionality.
Media Element provides some default skins and greater documentation for building your own player designs. You can also try building a music/audio player out of the same codes. Both scripts provide documentation along with an API and they’re both fantastic libraries. Keep these in mind when you need self-hosted videos on any future projects.