Specialized Sequoia Elite/Motobecane Fantom Cross Team Ti/'85 Trek 520
Need a LOT more info on this and PICTURES! When you want to completely remove a rear derailleur, you generally have to use a Chain Tool and break the chain so you can remove the chain from the derailleur.
More details and pictures and let us know exactly what you have.
I'll guess that the hanger broke off the frame and the derailleur is hanging free on the chain.
If you're lucky, your frame has a replaceable hanger, but the derailleur itself may be bent -- perhaps beyond salvage -- and the hanger alignment on the frame may need adjustment. Unless you have the tools and skill to undertake this job,your best bet is to take your bike to your LBS and have them assess the damage.
It's as JohnD suggested, the hanger has broken. you need to get a replacement, would either go to the shop you got it from, or any good LBS, and purchase a replacement hanger (they are frame specific), fit and you will have a working bike again.
The dropout looks fine, the hanger which is a seperate part has broken (not the derailleur, this will be ok), this is a part which is designed to fail, so that the frame isn't damaged (works most of the time), meaning the cost of repair is only for the hanger, not the complete frame.