Aside from docker login, Docker isn't "connected to a registry" per se. Registry names are part of the image name, and Docker will connect to a registry server if it needs to pull an image.
As a specific example, the official Docker image for Elasticsearch is on a non-default registry run by Elastic. The example in that documentation is
docker pull docker.elastic.co/elasticsearch/elasticsearch:7.17.0
# ^^^^^^^^^^^^^^^^^
# registry host name
You don't need to otherwise configure your system to connect to that registry, download an index, or anything else. In fact, you don't even need this docker pull command; if you directly docker run the image, Docker will download it if it doesn't have a copy locally.
The default registry is Docker Hub, docker.io, and this cannot be changed.
There are several alternate registries out there. The various public-cloud providers each have their own, and there are also several free-standing image registries. Each has its own instructions on how to set it up. You always need to include the registry name as part of the image name. The Google Container Registry has a simple name syntax, for example, so if you use GCR then you can
# build an image locally, labeled to be stored in GCR
# (this step does not contact or use GCR at all)
docker build gcr.io/my-name/my-image:tag
# authenticate to the registry
# (normally GCR has a Google-specific login sequence)
docker login https://gcr.io
# push the image
docker push gcr.io/my-name/my-image:tag
# run the image, pulling it if not present
docker run ... gcr.io/my-name/my-image:tag
