Tiny copper waffle squares inside computer
They are heat sinks. You definitely should put them back on, as they are there for a reason. If the chips below it get too hot, they could be damaged.
I would purchase some thermal adhesive to reapply them. Do not buy thermal paste, as that will not bond them back together. Be sure to clean the heat sinks and the chips with some rubbing alcohol to remove old adhesive and dirt.
These are heat sinks and must be in place for reliable operation.
The shape of those copper devices is designed to dissipate heat effectively from the part on which they are placed. The "waffle" shape is intended to increase surface area, making it easier to remove heat from the device.
The heat sinks in question are responsible for cooling SMD inductors (coils) that form part of the CPU voltage regulator module. Without the heat sink, they could overheat when the CPU is under sustained load, which can reduce the service life of the system or cause it to fail altogether.
Use thermal adhesive to secure the heat sinks back onto the inductors. As Keltari mentioned, be sure they're clean before applying the adhesive; poor contact between the inductor and heat sink can reduce cooling effectiveness.
These are indeed heatsinks, but in contrast to what everyone else has said, these are part of a bolt-on aftermarket up-sell modification "upgrade" offered by the laptop distributor, XoticPC, and are for all intents and purposes largely useless. (See : Copper Cooling Upgrade)
The cheap quality of adhesive they used to sticker these things on (wherever they originally put them) should be a testament to just how much nothing they were actually doing for you. Clearly, they've fallen off completely and you didn't even notice. Luckily they didn't manage to damage anything or short out your motherboard while they were rattling around loose inside your case.
You probably opted-in to this upgrade when you purchased the laptop. If anything, I'd get in contact with Xotic and try to get your $60 back. Throw those copper things away, or keep them on your desk as ornaments if you like. You don't need them in your laptop.
As others have already responded, the blocks are heatsinks. But I don't believe the location in the image is where they are supposed to go. In fact, I don't think they're normally supposed to be in there at all! As you can see in other images of the GE60 motherboard, the waffles are nowhere to be found:
Did you happen to buy your laptop used? If so, you may have ended up with a retrofitted system, for example using the process outlined on this overclocking thread:
"...As of right now all I did was put two low-profile heatsinks on top of the GPU heatsink..."
The heat sinks are attached to the chip so that they absorb the heat. The air between the waffles cools down the absorbed heat; as a result, a constant temperature is maintained.