GIMP is different from Photoshop, but also similar in many ways. One of these differences is its behaviour when you copy and paste. Pay particular attention to step 3 below - it's different from Photoshop!
Make a selection
Copy Ctrl+C, then Paste Ctrl+V
In the layers panel hit the New Layer icon to anchor the floating selection to its own layer, or alternatively click Layer > To New Layer Shift+Ctrl+N
Move it with the Move Tool M
Note: Use Command (for Mac) instead of Ctrl (for Windows/Linux)
If you are wondering about the "floating selection" behaviour, there was a time Photoshop had similar behaviour, way back before it had support for layers, but it no longer exists in Photoshop. It still exists in GIMP, and can in fact be used to move stuff around without using layers - a hangover from the early days of image editing software if you like.