Do games for some PS versions work for other PS versions?

E.g. A game was made in the era of PS2. But then, you get a PS4. Can you play the PS2 game on PS4 or a PS4 game on PS2?

P.S. The PlayStation versions were just an example and purely coincidental. Please consider all PlayStations in your answer.

It depends.

Each console generation uses different hardware, which behave differently. Code written for one hardware will not necessarily work for another. This is especially true for consoles, which usually make heavy changes to their hardware from one generation to the next, usually removing old, obsolete features that can be achieved differently.

Older PS models

As a general rule of thumb, an older console will not run games made for a newer console. So in your example, no, a PS2 will never run a PS4 game. It just doesn't have the hardware to do it (it's not like Sony had a time machine that allowed them to include PS4 hardware in a PS2).

The other way around is more complicated.

PS2

In the past, newer consoles shipped with hardware used in older consoles in order to provide backward compatibility. For example, the PS2 had PS1 hardware, with which it ran PS1 games.

PS3

The PS3 used to have PS2 hardware at first. Then, some of that hardware was removed to reduce cost. At that point, PS2 games had to be partially emulated to run on the PS3. Finally, the PS3 shipped without PS2 hardware. Sony eventually released an emulator that allowed some select few PS2 games to run on all PS3 models. These games could be bought for the PS3 on the PlayStation Network.

Fortunately, all PS3 models are compatible to PS1 games through the use of emulation.

PS4

The PS4 only plays PS4 games.

While you can play PS3 games on PS4 using a PS Now subscription, the games don't run on the PS4 itself. Instead, they run on some servers in the cloud. Input is streamed from the PS4 to those servers, and image and sound is streamed back from the servers to the PS4. This also works with other devices, such as a PC.

A few PS2 games are available on the PlayStation Network and can be played on the PS4 via emulation. The original PS2 discs are not compatible to the PS4, however.

PS5

The PS5 supports running PS4 games. According to pocket-lint.com, this is achieved through emulation:

As with the Xbox Series X and S, backward compatibility is performed through emulation, which means some features and/or games may not work or may exhibit some bugs.

There is currently no information on compatibility with PS3 or older games.

Emulation is a technique that translates code written for one machine, so it can run on another. Obviously, running a game through an emulator requires more power than running the game directly, hence why it's used exclusively by newer (more powerful) consoles to run games written for older (less powerful) ones. A PS2 could never hope to play PS4 games through emulation.