Cell division is the process by which one cell becomes two. If we consider multicellular fungi, cell division happens whenever the fungi grows by one more cell. Cell division allows the growth of the organism. Cell division does not necessarily contribute into making a new multicellular organism.
Fragmentation is a type of non-sexual reproduction. An organism gets split ("intentionally" or "unintentionally") and the different fragments develop into independent mature organisms. Sometimes, on fragment is small than the other one, so that one fragment can be seen as the "parent" and the other one as the "offspring".
Budding is a type of reproduction by fragmentation, where the splitting occurs due to a specific developmental pathway and not through some external environmental cause (such as getting cut in two by a researcher's knife, a falling tree or the bite of a predator). In other words, in budding, the splitting is "intentional".
P.S. You won't need to know this biology stuff to get a good score at your TOEFL. On an unrelated personal note, I found this exam particularly unpleasant when I had to take it. Especially when we have to converse with a microphone while somehow ignoring the accents and english mistakes your neighbours are making while talking into their respective microphones. Good luck for your exam by the way :)