[HTC One S] MTKnife:
UMS (I typed "UMB" by accident before--I've just corrected it) stands for "USB Mass Storage" or "Universal Mass Storage", though the technical name for the standard is "Mass Storage Class" ("MSC"). This is the traditional standard for detachable devices with storage, such as external hard drives, flash drives, and SDcards, and it's supported by every operating system. It's also been used for MP3 players, digital cameras, and phones.

Here's the Wikipedia article for more info: http://en.wikipedia.org/wiki/USB_mass_storage_device_class

MTP stands for "Media Transfer Protocol". It was designed originally (by Microsoft) for copying music to and from media players. One advantage is that it maintains an index of the whole file system, which means it doesn't have to rescan everything every time something's modified (that's a big advantage for media players); I once had an MP3 player that could use either UMS or MTP, but when you used UMS, you couldn't sort your songs by their MP3 tags, which probably had something to do with that index as well (newer devices have more sophisticated media player apps that aren't hampered by this limitation).

However, the big difference between MTP and UMS is that an MTP storage device (like the internal storage on your phone) doesn't "mount" on a host device (like your computer) like an ordinary disk drive. This limits what you can do with the storage device, which can be annoying, but also makes it more "idiot proof": the computer can't corrupt the file system, and you won't damage it by pulling the USB cord out while the computer is accessing the storage, and I think it even means that the computer and the phone can access the storage at the same time (I don't have an MTP device here at the moment, and so I can't test that.)

Here's the Wikipedia article for more info: http://en.wikipedia.org/wiki/Media_Transfer_Protocol

Why do people get worked about the difference? Well, the main reason is that, while an MTP device is great for dealing with music and other media files (assuming you're using a program designed to use it), it's a PitA when you want to use your phone (or MP3 player) to transfer and carry around files, because you can't read files directly from it or save files directly to it from apps not designed to use MTP (including, say, the Windows command-line interface); pretty much the only way you can interact with it, in fact, is to drag and drop (in Windows 7, for example, it shows up under "Media Devices" rather than "Devices with Removable Storage"). That means, for example, that if I write a report in DOC format at home and save it onto my phone's internal storage, I can't plug the phone into my computer at work, double-click to open it up, and then save the document when I'm finished editing it--I first have to transfer it to my work computer's hard drive, and then I have to transfer the finished document back to the phone. When it comes to rooted phones, custom recoveries can't mount an MTP partition (I think this is correct, but maybe someone has written a recovery with a way to get around the problem), meaning that you have to turn the phone on and drag and drop to get files onto it, which can be annoying when you decide while in recovery that you want to flash something you hadn't planned to flash earlier. MTP files transfers are also slower than UMS transfers.

I love HTC phones, but when it comes time for a new one in May, if HTC is offering only MTP phones and the competition is offering UMS, I'm not going to be buying HTC.