There are increasing numbers of people using touch devices like tablets and phones. Recently I've run into a surprising number of problems using a tablet to access websites of large businesses - you'd assume they could afford qualified web developers. Websites can be programmed to detect if you are using a desktop or mobile device and also different operating systems. But you also have to design a website for accessability controls used by people with disabilities, And also the use of "touch" devices instead of a mouse or keyboard.
On a touch device, you cannot use the mouse "hover" function to display text, change an image, or gather data. Information you might want to display by hovering cannot be seen on a touch device. If the information is important, the web designer needs to find a duplicate or even a different method. Or just forget you ever heard of "hover".
Mobile v.s. Desktop
My tablets has a 10" screen that can display a desktop formatted screen, yet it is identified as a "mobile" device. The desktop version of the page is usually far superiour - mobile pages often leave out a lot of the information you get on a desktop page, making it an annoying touch-touch-touch navigation to get to what you want. (Similar to that annoying click-click-click.) I'm lucky to use Firefox Beta that has an addon to force "desktop" pages to be shown (you can uncheck it in the menu).
There are problems using a drop-down menu on a touch device. These menus are used by many large online sellers as as home-brew sites. The problem of trying to use these by touch is often made more difficult if a small font size is used. My rather small fingers often select the wrong item from these lists. I've started using a touch-pen (smudge-control) which helps choose the right item, but I really shouldn't have to. On a phone it's much worse of course, but then the website might send me a "mobile" page which works on small screens.
My favorite weather site started using "sliders" to control the sequential radar maps - speed, transparency, etc. - which are impossible to "slide" by touch. All I can do is tap an empty part of the slider bar (and sometimes there's not enough space to do that). Basically I have no fine control at all, so I end up stepping the maps manually. (They also continue into "future" maps with no way to turn that off, which is another usability problem altogether.)
Special Mobile Features
Many touch devices can detect a "hand wave" close to the device which can be used as an input to the operating system. Yet few apps for mobile devices make use of this unique feature. Web browsers on tablets should allow me to at least "swipe" to navigate to a previous or next page, since they seem to move up and down as you might expect. Ebook readers have the swipe navigation figured out. Luckily, I found a Firefox for Android addon that adds this functionality, but basic touch navigation should be built-in to mobile browsers from the start.