Cross-save means starting on one platform and continuing on another. Every major platform now supports it in some form, but the implementations differ in ways that catch people out, particularly around what transfers and what does not.
How the platforms differ
PlayStation ties progress to your PlayStation Network account. If your games are on the same PSN account, they sync automatically with no action needed, which makes this the smoothest of the three implementations.
Xbox does the same through Xbox Network. The main Xbox-specific caveat is that some games require the Xbox network to be the primary account on the console, which is a console-side setting rather than something the game controls.
Nintendo's approach is different because Switch saves are console-local. Progress transfer requires an explicit export and import through the game's own system, and Nintendo's own first-party titles handle this through a dedicated backup feature.
Steam, Epic, and GOG handle this through their own cloud save systems. The most common problem here is not the cloud sync itself but a game that stores save data in a way the platform does not support, which makes sync silently not work.
What usually does not transfer
Cosmetics and inventory transfer in most cases. What frequently does not transfer is anything bound to a competitive ranking system, and in some games anything tied to a platform-specific achievement list, since achievements are tracked per platform.
Some games also separate a cross-save from a cross-play profile. Cross-play means you can play with people on other platforms, which is unrelated to whether your own progress moves between them. A game can have one and not the other, and the store page will often say cross-play without mentioning cross-save.
Before you buy twice
Check the store page specifically for the word cross-save, not cross-play. Then check the publisher's support documentation, because the store page frequently omits it and the support page is authoritative.
If a game supports cross-save but you are switching between a platform with cloud save and one without, verify that your current platform actually implements cloud save for that specific game before you assume your progress is safe.