Termux (Linux terminal) is super cool, and you can install a lot of cools programs through it as well. Add in the plugins for tasker integration/etc and it can be used for some pretty cool things (ie, use wol to wake up your computer when you get close to your house, etc).
By default the app is just a terminal with an on-screen keyboard. Packages installed and run through it are normally terminal only, although there is some amount of support for connecting to a separate app that runs an x server.
There are several other projects that download arm Linux distributions, and install them inside termux to get a full desktop going though. Normally for these you actually interact with the desktop through a VNC client connected to a localhost VNC running on the Linux distro. A VNC client is a good way to handle accessing the desktop since most of them come with a lot of UI design/polish for controlling a desktop computer via a phone screen.