I have photos in my DropBox folder that came from a shared DropBox file. If I delete them from my computer they are deleted from my online DropBox folder. Will they also be deleted from the folder of the person who shared them with me?
Yes. When you share a folder with someone else it's synchronized between the two of you. If you delete something from the folder it will be deleted from their storage as well.
You should "Leave" the folder before you delete anything from it. Right-click on the folder and choose the Dropbox "Shared folder settings..." option. Then under "More Actions" choose "Leave folder".
More from Dropbox Help: How do I leave a shared folder?