I have a GitLab repo, which uses docker container registry.
I want to push some of built containers to production using only one button click.
I heard that Kubernetes and Docker Swarm can help me, but I don't understand what it is
Can you explain in human language what is docker swarm and how can I use it to solve my task?
I want to push some of built containers to production using only one button click
You don't need docker orchestration for that, just docker.
More specifically, docker image push
(previously docker push)
see "GitLab Using Docker Build"
Container orchestration is what is needed to transition from deploying containers individually on a single host, to deploying complex multi-container apps on many machines.
That is not what you are trying to do here: you just need to push images to a registry.