I can't seem to figure out what the problem is with our toilet. I'm living in an apartment and I don't want to bother landlord for something that I might be able to easily fix (I've bothered him quite often). Each time we flush we get a loud screeching noise I guess you can say. I've seen somewhere that if I lift the (black thing that floats) when flushing, if the noise stops the problem is the valve. The noise does affirmatively stop and when I let go to start making water go back in the tank it sounds perfect. Does this still mean I have to change the valve and if so what is this valve called?

That fill valve has a washer and that is the likley source of your sound, due to vibration. I would not advise you to do the repair unless you have done it before. The LL should cover this maintenance issue.