I have a client whose site was created in WordPress. I need to install a video player which will allow for multiple videos to be played, at the users option.
Any recommendations?

Check this plugin. I haven't understood completly what you need to do ( i mean if your video source is web or your own server) but i guess if this doesn't fit, you could find a pluing easy. There are many good plugins outhere

Sorry about the confusion. Yes, the video files would be hosted locally on the server. Just need a good video player to put it all together.

Why not upload to Youtube and install a Youtube plugin in in Wordpress?

S