I'm not sure these categories can really be cleanly applied to Citrix. Let me just explain in a nutshell how it works and you can be the judge yourself which of the (if either) it should be.
I have a farm of Citrix servers that I deploy WPF to. The servers are basically just Windows machines, so I can browse for files, upload, interact with the local file system in any way. The app itself to the Citrix server just like it was a personal computer. Citrix technology basically just transmits a picture of whatever apps each user has open on the server(s). It does this by the user installing a client (web browser plug-in), and all that comes across the wire is the compressed graphics info. There is no discernible lag, so it's basically just like I'm working directly from the server. I can't copy objects directly to my laptop from these web servers, because the OS I'm on there isn't really the same OS (although can browse through the network to my laptop and copy it that way very quickly).
That is Xenapp. I assume XenDesktop is the same as what we call 'Remote Desktop, but double-check me on that. This is what I use to log into a computer in the office from my home and control it. It works very much like the above except that, instead of logging into a server , it's used to log into a desktop PC.
Both technologies just transmit a (compressed) image, and both allow you to send keystrokes and mouse movements so that it's like you're working directly on that machine. As I understand it, Citrix is one of the few games in town with this kind of technology, and last I heard, even MS licensed it from them.
The typical usage is to install fat client apps on a Citrix farm so that they then become web/browser accessible from outside the work place. The apps are published on a gateway web site with links to the individual apps (although you can also browse through the file system and open that way). The only thing the user needs to have installed to do this is the Citrix client to decipher the visual stream. The client is free and lightweight.
So basically, I would say Citrix technology allows for fat clients to be installed on the Citrix server and then accessed like thin clients.
There are a few key differences between Citrix deployment and the way typical web app works. One is that the user has to actually close the app out, not just their local web browser , otherwise the app stays running on the Citrix server. By default that doesn't typically happen because from the Portal, a particular app will be published, so that only that particular app pops up on click of the link (not a desktop or Windows Explorer). So when the close the 'picture' of it running in the browser, they do so by closing the 'X' on the app. But if they're crafty, they're can disconnect the client from the server and leave it running. That can be handy if ones need to some work that shutting downt the laptop would otherwise close out (long-running datawarehouse pulls, etc). Another difference is that speed and performance are pretty much the same no matter the location of the user(at least with XenaPP). Normally, if you have a Wide Area Network, and you say, deploy an ASP.NET web page on web server in City A, the user in City B 1000 miles away may have a bit of a lag, since the web app may have to query a database server, then spit out some Javascript, that then gets consumed and ran on the client. With Citrix Xenapp, everything is occurring on the server in City A. In Citry B, the user is just getting a compressed picture stream. For this reason, it's better to avoid too fancy graphics, because they waste bandwidth and usually get autocompressed to look weird anyhow. But assuming that is done and the farm doesn't suck, performance will be appreciably the same in India or the Philipines or the United States for the same app. Another difference is that the data is inherently Sandboxed, and there are is no URL unless you decide to put the app on a web server and then have users access it through Citrix (which I've seen done in companies with sensitive data using offshore vendors because of the Sandboxing and speed benefits). But if you do that, you have to open the web app from within the Citrix portal and then you can run the browser on that server (you can't just put a link to that web app from the web). Finally--and maybe this is just where I work--but the load balancing seems to work a little differently than with web servers. Users tend to get thrown on the same server if they already have another app open. That can be handy for copying files, etc, but also means less balance in the load for particular servers at times, so that you typically don't want the overall average load to go high (need more servers).
Hopefully that helps explain it and give you an idea. Citrix just sends a picture of the wire that you can use to remote-control a machine. I would say it's kind of "both" on the thick or thin client question. Typically it is used to deploy Winforms, WPF or other 'fat client' technologies, and is largely unnecessary for technologies that already allow for thin clients (web apps). But sometimes web apps are pushed through there also, for various reasons.
