Re-glue plastic edge banding on a desk
Laminate is usually held on with contact cement similar to this



This type of adhesive is spread on both surfaces to be joined, allowed to dry and then pressed together. Pressing firmly with a small wooden roller (or a rolling pin) helps make the bond tight.

It is important to make sure the parts are perfectly aligned before the surfaces touch because they cannot be moved once they contact. This is often done by placing a piece of very smooth paper between the two surfaces while aligning, then sliding it out. 

If the band is already off along much of its length, it is better to remove it fully to make it easier to apply the glue. Be careful when prying off to make sure that it does not crack. if it does not come off easily, wedge it away from the base with a pencil or something similar (a chopstick) to give you room to get the glue on both surfaces. Again, be sure to let the glue dry before joining. How long it needs to dry will be explained on the packaging.

With contact cement, no clamping is needed, because it achieves full strength almost immediately.