As Pilot6 says, when you install Ubuntu using the graphical installer in a live environment, the installer you are using is Ubiquity. The desktop ISO images, which are the most popular way to install Ubuntu, use Ubiquity. (See also, What is ubiquity?.)
In contrast, when you install Ubuntu using the text-based server, minimal, or -- for older releases that had them -- the alternate ISO image, that uses the Debian installer. More precisely, it is "d-i," which is the Debian installer with the necessary changes for Ubuntu.
As the Installer/Development page on the Ubuntu wiki (by "Contributors to the Ubuntu documentation wiki") says:
Ubuntu and its various flavours (Kubuntu, Edubuntu, Xubuntu) have two
installers. One is a modified version of the Debian installer,
known as "d-i", and is written in POSIX shell script and C. The other,
Ubiquity, is used on our live CD (or "desktop CD"), features a
from-scratch user interface design, and has a good deal of front-end
code written in Python, but behind the scenes still uses d-i code for
many back-end tasks where it's important that we don't end up
maintaining two implementations.