I've been confused by the concept "forward and backward compatibility" for a long time. In Chinese, sometimes forward (前) and backward (后) represent different meanings under varying scenarios. E.g., "once upon a time" (从前) and "looking forward" (向前看) both contain the character 「前」, but their meanings are completely opposite. Similarly, the character 「后」 in "from now on" (今后, 此后) and "looking backward" (向后看) has very different meanings. Thus, when it comes to forward and backward compatibility, it's better to understand them totally in English, rather than translating into Chinese.
Compatibility from the developer's perspective.
The above figure demonstrates the forward and backward compatibility, note that in following dicussions, we always stand in the point of developer at current timestamp (i.e., version B in 2025/04).
Backward compatibility
Suppose we are the developer of a software named X (as developers are the ones who need to care about compatibility issues at the most), and it has 3 releases in history (A in 2024/04, current B, and future version C in 2026/04).
We say the version B of X is backward compatible to version A , if the data/files in version A, can be handled in version B without errors.
Note that B is the future version of A, so backward compatibility represents the stability of a software (interface/protocol, etc.).
To examplity, Microsoft Word 2023 can still open .doc files (created by Word 1999, maybe), so it has a good backward compatibility.
The fewer breaking changes, the better backward compatibility.
Forward compatibility
Again, we still stand in the point of developer at version B in 2025/04. We say the version B of X is forward compatibility to version C, if data/files in version C, can be handled by B correctly. Compared to C, version B is the old version. So forward compatibility depicts the extendibility of a software, in other words, whether the developer of B can forcast its functionality for its future releases. Take a real example, NVIDIA maintains the forward compatibility of PTX, such that the binary compiled with PTX now can run on future generations of GPUs.
It's intuitive that maintaining the forward compatibility is much harder than the backward compatibility, as programmers can always "retain" the elder code to handle the past data. But for forward compatibility, developers should have an anticipatory design for the software, e.g., allocate some bits/fields in the protocol. Although some new features may not work in old version (B), but most functionalities should work, which is enough for most of softwares in the world.
Outra
When we talk about compatibility, we should always stand in the point from the perspective of developer (rather than users) at current version. If the data from past (backward) can still work in current version, it is backward compatible; if the future data can work in current version, it is forward compatible.