Depends on the blu-ray player. Some players have component or composite connections, which will allow you to hook up a Blu-ray player to a Standard or Enhanced Definition TV. For example, the PS3 some with a composite cable that will hook up to a standard definition TV, and a component cable is also available for enhanced definition TVs. It also has the HDMI port on the back for HDTVs, but only HDTVs use HDMI. If your player only supports HDMI, you might not be able to hook it up to your TV. You will be able to play both DVD and Blu-ray, but not at High Definition. While you will not get the higher resolution, you should still get better image and better sound (depending on your sound system), as well as enhanced interactivity and interconnectivity.
If you are planning to play games from something like a PS3 (the Wii does not do HD so you don't need to worry about it), I recommend getting an HD screen, since often the lettering is too small to be clearly seen on an SD screen, and only an HD screen will show the words clearly.