Sometimes I think German will make me go completely crazy. How can you have so many words for expressing "falling"?
Can someone explain me the difference and in which way are they mostly used?
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Fallen means to fall, to fall down. The generic meaning is to have a downwards movement, falling. If you throw a stone from a very high position, this stone fällt. You cannot interchange it with the other words.
Fallen has a lot of meanings that I'm not going to cover in this answer, because they aren't related to the question.
Just some brief examples:
Now, when a person fällt, he drops to the floor. When a pencil fällt, it drops to the floor.
In both cases, you can use hinfallen. In the first example, hinfallen doesn't really add much here. Hin- implies dropping to the floor. In the latter example, however, there's a subtle difference. You would use hinfallen especially in those cases where it falls accidentally to the floor. For example when you were holding that pencil in your hand and it slid out of your grasp. Fallen, again, could also imply the general process of falling. Think of that stone you threw. Thus, you'd prefer hinfallen over fallen if fallen were ambiguous. In case of this pencil, it doesn't really matter though.
For using umfallen there is one condition to be met. The object needs to be in an upright position. When I was writing my comment earlier, I was just thinking of a person. My comment was universally incorrect, because you wouldn't interchange umfallen with fallen or hinfallen when talking about a pencil.
You put it on the table, it rolls to the edge and falls down. It is gefallen or hingefallen (you'd prefer the former one here). If, however, you put it in an upright position and the pencil topples over and is still lying on the table, it is umgefallen.
Now consider a person, that has fallen down to the floor, you can use *fallen, hinfallen and umfallen without any difference in meaning (usually), provided the person was in an upright position. If the person has been fallen out of bed, he's just "aus dem Bett gefallen".
One last example:
A pile of books that toppled over is umgefallen or umgekippt. When you would say that something toppled over, use umfallen or kippen.
Stürzen is a synonym to fallen. There's also hinstürzen (quite rare, I guess) and umstürzen. You see that you can apply the same prefixes here again. And you can use stürzen in the two examples above with that king and the empire, too.
To cut a long story short:
All these have slightly different connotations.
While EM1 wrote that hinfallen can also be used if something is dropped (herunterfallen), this is not true in all dialects (there are, however, German dialects where both are equivalent).
Hinfallen is normally used when a person trips and falls. You could, for example, say:
Schau, da hinten ist ein Kind hingefallen
This is equivalent to stürzen in this meaning. If used in the same meaning as herunterfallen in some dialects, it is not equivalent to stürzen!
This word is used to say that something or someone that/who previously was in an upright position fell over and is now no longer in an upright position. For example:
Durch die Last des Schnees ist der Baum einfach umgefallen.
Er wurde ohnmächtig und ist einfach umgefallen.
These words are used when something falls from an elevated position, either by itself or by someone dropping it.
Mir ist der Stift runtergefallen.
Das Kind ist vom Klettergerüst runtergefallen.
This word means to fall in general (also some other meanings apply as EM1 correctly pointed out). Usually, you would use the specialized expressions above, as you can not say the following:
Mir ist der Stift gefallen.
In addition to what I wrote on fallen: As EM1 points out correctly in the comments, it is very well possible and very common to say something like
Mir ist der Stift vom Tisch/aus der Hand gefallen
Mir ist die Tasse auf den Boden gefallen
It would be very very unusual (I'd even say impossible) to use Mir ist der Stift gefallen without any attribution from where or whereto it fell.