I cut the mesh into pieces but it's still one object.

As I said in the title, I cut the mesh into smaller pieces but when I try to move 1 piece, it moves all of them.

Is there anything I can do or do I have to start over?

When using screenshots please whow the whole blender interface, you have cropped out vital information.
How did you cut the object up ? Did you separate the selection to make separate objects §
Do they show up as separate objects in the outliner ?
Have you selected the single object you want to move
If they are rally separate objects have you enabled proportional editing