What are they emulating that keeps them from just being terminals? What does a plain-old "terminal" look like, if they are just emulators?
A terminal is the end of a line. So, back in the day when the computer was a mainframe serving many user accounts, what you'd be sitting at with your keyboard and display would be a terminal.
A terminal emulator is when you're using a computer (a Turing machine) to provide the function of a terminal in software. This usage would typically come up because the computer would be 'imitating' a particular type of terminal in order to communicate with the mainframe.
Source and more explanations are in Terminal or terminal emulator?.
Until the 80's, there were no PCs, but mainframe terminals. These just had a command line interface with no GUI. The Linux terminal emulators we use today emulate the settings of those old terminals.
Mainly for historical reasons. In the mid 1980s, when terminal emulators were invented, they were called "terminal emulators" to distinguish them from the "real thing." The term is ambiguous nowadays, because the "real thing" is obsolete and only found in very old installations or museums. See terminal emulator and computer terminal.
(Credit: Flickr user Jason Scott)