Ps4 – download a game on 2 different PS4

As TZHX already wrote in a comment: The PS4 is not backwards compatible, meaning you can only play Playstation 4 games on it. The only streaming capability it has is that you can stream PS4 games to a PS Vita. In the future, it will support streaming games of the PS3 and older platforms online via Playstation Now with a rental system. Only two days ago it went into open beta.

When you open the PS store on the PS4, only PS4 titles appear, so you cannot download PS1, PS2 or PS3 games at all. The only PS3 games that appear are PS Plus titles that you can claim for later download on a PS3, but cannot download on the PS4 itself.

The PS1 games available for download in the Playstation Store can only be played on PS3, Playstation Portable (PSP) and PS Vita, while the PS2 games from the store can only be played on a PS3.

Not all PS1 games are available to download on Vita, and some of them are region specific. Some also work when they're on the Vita but cannot be downloaded directly from the store.

When you access the store on console or at store.sonyentertainmentnetwork.com you can see which platforms are supported by each game.

As of right now (March 2016), you can have two Primary active PS3 systems and one Primary active PS4 system according to the PSN FAQ page.

You can sign into additional devices and download your games to play while logged in, but as soon as you log out the games become unlicensed for play by other users. I'm not sure how that works with concurrency though... eg, can you play against your self between two systems? IDK.

Note, even this page isn't always current but it's a good guide. For instance you can have three activations across PSP, Vita, and PS TV, not the two listed.