! [rejected] main -> main (non-fast-forward) with the hint "the tip of your current branch is behind its remote counterpart" means the remote branch holds a commit your branch does not contain. Either someone pushed while you worked (run git pull --rebase, then push) or you rewrote a commit you had already pushed (push with --force-with-lease).

Run git status first and ask which of the two happened. Pulling after an amend or a rebase brings the old commit back next to the new one; force-pushing over a teammate's work deletes it. The right command depends on the cause, not on the message.

The error

PS> git push origin main
To C:/.../a2remote.git
 ! [rejected]        main -> main (non-fast-forward)
error: failed to push some refs to 'C:/.../a2remote.git'
hint: Updates were rejected because the tip of your current branch is behind
hint: its remote counterpart. If you want to integrate the remote changes,
hint: use 'git pull' before pushing again.
hint: See the 'Note about fast-forwards' in 'git push --help' for details.

The long absolute paths were shortened; nothing else changed.

Why it happens

A plain push may only move a remote branch forward: the commit the remote's main points at must already be in the history you send. Git says non-fast-forward (rather than fetch first) when your repository already has that remote commit, usually because you ran git fetch or an IDE fetched for you, and can see your branch does not include it. git status shows the shape of the problem in both cases: 

Your branch and 'origin/main' have diverged,
and have 1 and 1 different commits each, respectively.

Case one: a teammate pushed david change after you cloned, and you committed on the old tip. Case two: nobody else pushed; you ran git commit --amend on a commit that was already on the remote. Amending creates a new commit with a new id, so the pushed one is no longer in your branch, and Git prints exactly the same message.

The fix

Case one, someone else's commit is on the remote: bring it in, then push.

PS> git pull --rebase origin main
Successfully rebased and updated refs/heads/main.
PS> git push origin main
   ecded88..9bbee91  main -> main
PS> git log --oneline --graph
* 9bbee91 maria change
* ecded88 david change
* dbb247b first

Case two, you rewrote your own pushed commit: replace the remote tip, but safely.

PS> git commit --amend -m "maria change, reworded"
PS> git push --force-with-lease origin main
 + 9bbee91...af34640 main -> main (forced update)

--rebase replays your unpushed commit on top of the remote one, so the push becomes a plain move forward and the history stays a straight line. In case two, --force-with-lease overwrites the remote only if it still points where your last fetch saw it. In a separate test Maria pushed, fetched and amended her commit; David then pulled and pushed a new commit; Maria's lease push was refused instead of deleting his commit: 

PS> git push --force-with-lease origin main
 ! [rejected]        main -> main (stale info)

Fetch, look at what arrived, and decide again. Never force-push a shared branch others have already built on without telling them.

How it was reproduced

Git for Windows 2.45.2 on Windows 11, Windows PowerShell 5.1. A local bare repository served as the remote, with two clones (David Chen and Maria Garcia, fictional). David pushed a commit after Maria cloned; Maria committed, ran git fetch and pushed: rejected with exit code 1. git pull --rebase and a second push succeeded. Maria then amended her pushed commit; the push was rejected with the same message and --force-with-lease replaced it.

Frequently asked

What does non-fast-forward mean in git push?
The remote branch points at a commit your local branch does not contain, so moving it to your commit would drop that commit. Git refuses unless you integrate it first or force the push.
Should I use git pull or git push --force after non-fast-forward?
Pull (preferably git pull --rebase) when someone else pushed. Force with --force-with-lease only when you rewrote your own already-pushed commits with amend or rebase.
What is the difference between fetch first and non-fast-forward?
Both mean the remote has a commit you lack. fetch first means your repository has never downloaded it; non-fast-forward means it has, usually after git fetch, and can see your branch does not include it.

More decoded errors in the Fixes category. When the rejection says (fetch first) instead, see git push rejected (fetch first).