No, the Playstation 4 is not backwards compatible and you can only play Playstation 4 games on it.
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This depends on the model of your PlayStation three. Sony has issued out four different models of the PlayStation three, these are; The forty gigabyte, sixty gigabyte, eighty gigabyte and the one hundred and sixty gigabyte. The forty and eighty gigabyte were released first, but the forty gigabyte PlayStation three cannot play PlayStation two games. Any other model can.
You can not play PlayStation 3 games on a PlayStation 1, but you can play PlayStation 1 games on a PlayStation 3. All of the PlayStation consoles are backwards compatible.
You cannot play Playstation 4 games on a Playstation 4 console. You are also unable to play Playstation 3 games on the Playstation 4.
You can't play PlayStation games on the Internet, but if you have a PlayStation 3, you can go on the PlayStation network and play with other people through the Internet.
Playstation 3 can play old Playstation games, but Playstation 2 games are not compatible. You can load some playstation 2 games using the hard drive and an emulator, but not all.