What is docker.io in relation to docker-ce and docker-ee (now called "Mirantis Kubernetes Engine")?

Be wary of docker-ce

The accepted answer is under-complex.

docker-ce is provided by docker.com, docker.io is provided by Debian.

On the surface, this means you can install docker.io rightaway, while for docker-ce you have to attach an external repository from docker.com beforehands.

More importantly, however, although both packages provide properly released versions of Docker, they have a very different internal structure:

• docker.io does it the Debian (or Ubuntu) way: Each external dependency is a separate package that can and will be updated independently.
• docker-ce does it the Golang way: All dependencies are pulled into the source tree before the build and the whole thing forms one single package afterwards. So you always update docker with all its dependencies at once.

The problem with the latter approach is that it goes against much of what Debian/Ubuntu are trying to do.

If everybody did it the way docker-ce does...

...you would have 174 versions of many libraries on your system, which not only consume a lot of memory, they also make it essentially impossible to decide whether you have that version 7.6.5 of library XYZ with that horrible security vulnerability somewhere among them.
Let alone close that vulnerability (or all 109 instances of it you have).

Worse, one of the 174 versions is likely to be version 5.4.3 of XYZ as of three years ago, which had another, very different, but just as gaping security vulnerability that the world has long since forgotten about but that will still exist happily on your system.

Some remarks:

• Many web pages call docker.io "outdated". That is because it was unmaintained for about a year. As of August 2019, this is no longer the case.
• I learned all this today here and will now switch from using docker-ce to using docker.io -- and presumably never go back again.
• There is a reason why the Debian/Ubuntu packaging system is so complicated. A good reason.

Edit: As BobHy points out in a comment, the docker-ce approach also has an advantage: It is less likely to have compatibility issues with library XYZ. You have to trade off your risks.

Older versions of the Docker binary were called docker or docker-engine or docker-io

docker-io package is still the name used by Debian/Ubuntu for the docker release provided on their official repos.

docker-ce is a certified release provided directly by docker.com and can also be built from source.

Main reason for using the name docker-io on Debian/Ubuntu platform was to avoid a name conflict with docker system-tray binary.

http://manpages.ubuntu.com/manpages/precise/man1/docker.1.html

Docker has an enterprise version (EE) and a free community Edition version(CE)

Prior to installing Docker Community Edition (docker-ce from docker.com) you may need to remove older binaries.

Centos/RHL:

https://docs.docker.com/engine/installation/linux/docker-ce/centos/

sudo yum remove docker \
                  docker-client \
                  docker-client-latest \
                  docker-common \
                  docker-latest \
                  docker-latest-logrotate \
                  docker-logrotate \
                  docker-engine

Ubuntu/Debian:

https://docs.docker.com/engine/installation/linux/docker-ce/ubuntu/

$ sudo apt-get remove docker docker-engine docker.io containerd runc

Dry-run comparison on ubuntu:

$ sudo apt-get install docker.io --dry-run
Reading package lists... Done
Building dependency tree       
Reading state information... Done
The following additional packages will be installed:
  bridge-utils cgroupfs-mount containerd pigz runc ubuntu-fan
Suggested packages:
  ifupdown aufs-tools debootstrap docker-doc rinse zfs-fuse | zfsutils
The following NEW packages will be installed:
  bridge-utils cgroupfs-mount containerd docker.io pigz runc ubuntu-fan
0 upgraded, 7 newly installed, 0 to remove and 70 not upgraded.
Inst pigz (2.4-1 Ubuntu:18.04/bionic [amd64])
Inst bridge-utils (1.5-15ubuntu1 Ubuntu:18.04/bionic [amd64])
Inst cgroupfs-mount (1.4 Ubuntu:18.04/bionic [all])
Inst runc (1.0.0~rc7+git20190403.029124da-0ubuntu1~18.04.2 Ubuntu:18.04/bionic-updates, Ubuntu:18.04/bionic-security [amd64])
Inst containerd (1.2.6-0ubuntu1~18.04.2 Ubuntu:18.04/bionic-updates, Ubuntu:18.04/bionic-security [amd64])
Inst docker.io (18.09.7-0ubuntu1~18.04.4 Ubuntu:18.04/bionic-updates, Ubuntu:18.04/bionic-security [amd64])
Inst ubuntu-fan (0.12.10 Ubuntu:18.04/bionic [all])
Conf pigz (2.4-1 Ubuntu:18.04/bionic [amd64])
Conf bridge-utils (1.5-15ubuntu1 Ubuntu:18.04/bionic [amd64])
Conf cgroupfs-mount (1.4 Ubuntu:18.04/bionic [all])
Conf runc (1.0.0~rc7+git20190403.029124da-0ubuntu1~18.04.2 Ubuntu:18.04/bionic-updates, Ubuntu:18.04/bionic-security [amd64])
Conf containerd (1.2.6-0ubuntu1~18.04.2 Ubuntu:18.04/bionic-updates, Ubuntu:18.04/bionic-security [amd64])
Conf docker.io (18.09.7-0ubuntu1~18.04.4 Ubuntu:18.04/bionic-updates, Ubuntu:18.04/bionic-security [amd64])
Conf ubuntu-fan (0.12.10 Ubuntu:18.04/bionic [all])

$ sudo apt-get install docker-ce --dry-run
Reading package lists... Done
Building dependency tree       
Reading state information... Done
The following additional packages will be installed:
  aufs-tools cgroupfs-mount containerd.io docker-ce-cli libltdl7 pigz
The following NEW packages will be installed:
  aufs-tools cgroupfs-mount containerd.io docker-ce docker-ce-cli libltdl7 pigz
0 upgraded, 7 newly installed, 0 to remove and 70 not upgraded.
Inst pigz (2.4-1 Ubuntu:18.04/bionic [amd64])
Inst aufs-tools (1:4.9+20170918-1ubuntu1 Ubuntu:18.04/bionic [amd64])
Inst cgroupfs-mount (1.4 Ubuntu:18.04/bionic [all])
Inst containerd.io (1.2.10-3 Docker CE:bionic [amd64])
Inst docker-ce-cli (5:19.03.5~3-0~ubuntu-bionic Docker CE:bionic [amd64])
Inst docker-ce (5:19.03.5~3-0~ubuntu-bionic Docker CE:bionic [amd64])
Inst libltdl7 (2.4.6-2 Ubuntu:18.04/bionic [amd64])
Conf pigz (2.4-1 Ubuntu:18.04/bionic [amd64])
Conf aufs-tools (1:4.9+20170918-1ubuntu1 Ubuntu:18.04/bionic [amd64])
Conf cgroupfs-mount (1.4 Ubuntu:18.04/bionic [all])
Conf containerd.io (1.2.10-3 Docker CE:bionic [amd64])
Conf docker-ce-cli (5:19.03.5~3-0~ubuntu-bionic Docker CE:bionic [amd64])
Conf docker-ce (5:19.03.5~3-0~ubuntu-bionic Docker CE:bionic [amd64])
Conf libltdl7 (2.4.6-2 Ubuntu:18.04/bionic [amd64])

The docker-ce binaries will tend to be the latest versions and include docker-ce-cli.