My Arduino started to behave strangly, I couldn't upload programmes anymore and a Ledbar I had attached stayed on (while it should be of according to my program).
By some inspection I saw this:
Which part is this? Is my Arduino broken?
Yeah It looks like you did. That part that is cracked in half looks to be the Diecimila's reverse protection diode. It is there to protect the main circuits from the power supply being connected backwards. My guess is you must have applied a reverse voltage for a good amount of time before that diode got to that point. I can't tell you what kind of other damage you did, but I'm fairly sure that is the reverse protection diode. I'm surprised it's working at all now, at least from external supplies. USB power should still work.
That is quite a spectacular component failure you got there.
Wait for confirmation by someone else before taking this post for anything.
It is actually an arduino from my university. I only connected it via USB when I used it. It worked for a few weeks.
But before, it was certainly used in combination with an external battery by someone else.
It is indeed an Arduino Duemilanove 
ANyhow it is not a reallly big problem, I got another one 
Just wondering whether it would be possible that I blew it up or not, and if so: how.
Most likely it got blown up by having some metal on the board short out to something. That would cause a large current to flow through the diode (forward, not reverse) causing it to heat up and crack.
It could be you placed the board on something metallic like a small bit of wire, test lead, etc. I've seen students place a board on a pair of scissors Not good results.
It's worth replacing that component and trying to see if the rest of the board is OK.
Or likely, someone was using the Vin pin to power some external component and drew too much current from the external power source.
Lefty
I had a waveshield and a custom printboard with 5 rgb led's attached to it. For a minute, I connected it to a quite big sennheiser headphone.
For a minute, I connected it to a quite big sennheiser headphone.
Connected to what?