You'll need an HD Capable TV to Play PlayStation 4
As technology evolves, there are some steps that are needed to move the masses forward. For example, we went from video and cassette tapes to CDs and DVDs, and before that we had vinyl. We're currently seeing a shift from DVD to Blu-ray as well, and now, on top of that, a lot of the world has started to move to 'HD only.'
This is certainly the case when regarding the PlayStation 4- the next gen console will only have HDMI adaptively, so if you want to make the most of it, you'll need an HD capable TV.
Luckily though, if you've bought a TV in the last 6 or 7 years it's more than likely your TV already has HDMI input, and if it doesn't perhaps it's time for you to update your living room a little.
Although it's not confirmed, it wouldn't surprise me if the Xbox One had the same connection options. We also don't know yet whether you'll be able to use an adapter, and if you can, what quality will be reached by using such equipment.