What type of DVD disk do you need to burn movies and play them on a DVD player?

Pretty much any "Name Brand" DVD-R should work.
Just about every DVD player has "suggestions" on which brand for the best use. They've usually just tested those to make sure they work or they're just the name of the manufacturer's DVD's they want you to use.
I believe the "Encoding" is the biggest thing on HOW you record the movie. Check the DVD player's manual to see what they can play.