Due to the hardware architecture, Playstation 3 games cannot be directly played on the Playstation 4, but there are methods that will allow PS3 games to be played on the PS4.
The Cross buy program allows you to purchase a single copy of a game for a standard price and will give you access to this game on multiple platforms. Typically cross buy games are on PS3, PS4 and PS Vita. This means if you buy a cross buy game, you own it on all 3 systems. Cross buy usually applies to digital copies of the game, but occationally retail versions of games will contain a digital download of the game for another system.
Just like with Playstation 2 game on the Playstation 3, developers and publishers have been releasing HD remakes or “Remasters”, as they are typically referred to on the PS4. These remasters are PS3 games that have been upgraded to have better visuals and take advantage of the increased power of the newer consoles. These releases are full retail releases and will cost money to by even if you already own the game on the PS3. The advantage of these releases are upgraded visuals, new trophies, additional content such as DLC and most often, increased performance over the past generation versions.
PlayStation Now is an upcoming service from Sony that will allow PS3 games to be played on the PS4 using a network connection. The service is similar to Netflix, but works by streaming video games over the internet rather than movies. The service is not yet available worldwide, but will soon be available in North America. Once available you will be able to stream PS3 games to your PS4.