Moodle is running on a server.
No, you can't install it on a smartphone. When Moodle is installed on a server you can build courses with your smartphone. For editing courses you need to use the web browser. With the mobile app you can get the student view of the course.
I found a very nice idea to get a really mobile Moodle. It's the MoodleBox running on a Raspberry Pi 3. The complete server is in a very small box. It provides its own wifi. You can use the MoodleBox with 15 or 20 clients. Take a powerbank and get the electric power for the MoodleBox. https://moodlebox.net/en/