Three numbers with meaning: major, minor, patch. The meaning is the contract, and the contract is what a tool compares when it decides whether an upgrade is safe. Choosing which number to move is a decision about what a consumer may assume.
About 1 min of reading
Choose the number.
A bug fix
Before you run it, say out loud what it will print. Then open the answer.
A new command
Before you run it, say out loud what it will print. Then open the answer.
Read the first example one line at a time. A line that prints shows what appears on screen; the rest set things up quietly.
Moving major sets minor and patch to zero. A version 3.4.2 released as a major is a mistake, because the number then claims compatibility that a major does not promise.
A version can carry a suffix for a build that is not final. That suffix is what keeps a release candidate from being mistaken for a release, which is a real confusion in a team that ships.
2.4.1 to 2.4.2 is fixes only. A consumer that pinned 2.4 can take 2.4.2 without a test, because the contract says nothing was added or removed.
2.4 to 2.5 adds features and keeps the old ones working. A consumer can upgrade and its existing calls still work, which is the promise the minor number carries.
2.x to 3.x allows removals and changes. A consumer that auto takes majors will get a broken program, which is why dependency ranges stop at the major.
Each one is defined here on this site. Open the entry for the full definition and the other words it points at.