No, you don't need Windows Media Center to watch DVD movies. While I'm not sure if Windows Media Player will play them natively, there are other programs available for free, online (VLC comes to mind) that you can use.
You can use WMP, but if you want to watch DVDs you'll need to install a codec pack like k-lite, or install VLC player, as Microsoft removed the DVD-playback codec from 8/8.1.
No, you don't need Windows Media Center to watch DVD movies. While I'm not sure if Windows Media Player will play them natively, there are other programs available for free, online (VLC comes to mind) that you can use.