For a while, X terminals were a reasonably popular way to give people comparatively inexpensive X desktops. These X terminals relied on X's network transparency so that only the X server had to run on the X terminal itself, with all of your terminal windows and other programs running on a server somewhere and just displaying on the X terminal. For a long time, using a big server and a lab full of X terminals was significantly cheaper than setting up a lab full of actual workstations (until inexpensive and capable PCs showed up). Given that X started with network transparency and X terminals are so obvious, you might be surprised to find out that X didn't start with them.
In the early days, X ran on workstations. Some of them were diskless workstations, and on some of them (especially the diskless ones), you would log in to a server somewhere to do a lot of your more heavy duty work. But they were full workstations, with a full local Unix environment and you expected to run your window manager and other programs locally even if you did your real work on servers. Although probably some people who had underpowered workstations sitting around experimented with only running the X server locally, with everything else done remotely (except perhaps the window manager).
The first X terminals arrived only once X was reasonably well established as the successful cross-vendor Unix windowing system. NCD, who I suspect were among the first people to make an X terminal, was founded only in 1987 and of course didn't immediately ship a product (it may have shipped its first product in 1989). One indication of the delay in X terminals is that XDM was only released with X11R3, in October of 1988. You technically didn't need XDM to have an X terminal, but it made life much easier, so its late arrival is a sign that X terminals didn't arrive much before then.
(It's quite possible that the possibility for an 'X terminal' was on people's minds even in the early days of X. The Bell Labs Blit was a 'graphical terminal' that had papers written and published about it sometime in 1983 or 1984, and the Blit was definitely known in various universities and so on. Bell Labs even gave people a few of them, which is part of how I wound up using one for a while. Sadly I'm not sure what happened to it in the end, although by now it would probably be a historical artifact.)
(This entry was prompted by a comment on a recent entry of mine.)
PS: A number of people seem to have introduced X terminals in 1989; I didn't spot any in 1988 or earlier.
Sidebar: Using an X terminal without XDM
If you didn't have XDM available or didn't want to have to rely on it, you could give your X terminal the ability to open up a local terminal window that ran a telnet client. To start up an X environment, people would telnet into their local server, set $DISPLAY (or have it automatically set by the site's login scripts), and start at least their window manager by hand. This required your X terminal to not use any access control (at least when you were doing the telnet thing), but strong access control wasn't exactly an X terminal feature in the first place.