Docker comes in two flavours: The Comunity Edition (CE) and the Enterprise Edition (EE). See this question for the differences. Just take Docker CE if you don’t know which to take.
There are two ways of installing it (thank you, Katu and Suor!). The official docker way is a bit more recent.
(A) Official Ubuntu Repositories
$ sudo apt-get install docker.io
In the past this way was discouraged as the docker package was super outdated. The universe sources are fairly recent now.
(B) Official Docker Way
The Ubuntu installation instructions list all you need in detail, but in most cases it boils down to:
(1) Set up the docker repository
sudo apt-get update sudo apt-get install apt-transport-https ca-certificates curl software-properties-common curl -fsSL https://download.docker.com/linux/ubuntu/gpg | sudo apt-key add - sudo add-apt-repository "deb [arch=amd64] https://download.docker.com/linux/ubuntu $(lsb_release -cs) stable"
(2) Install Docker CE
sudo apt-get update sudo apt-get install docker-ce
(3) Verify the installation
sudo docker run hello-world
The latest version can be found here. It was 18.06 in November 2018. This version is available in the sources from above, whereas the default Ubuntu package sources only contain Docker 1.5 (source). You can find changes in the docker release notes. Version 1.5 is so old that I could not find out how old it is.