Summerhouse door hinge repair/replacement
You're looking at a screw in, lift off  barrel pin hinge. 



The one shown above has a set screw to prevent unnecessary lift off, yours might not have it. 

There are better hinges out there for door and windows that are more available so look around. These hinges are going to be hard to find at your regular home stores. You'll need a speciality store unless you can reuse the ones you have and just mount them in new holes.

Spray some liquid wrench in them and then work them out... you might have to get rough...