Liithium-Ion polymer batteries are in the form of a soft pouch, often surrounded by a hard case for extra protection, though devices with non-removable batteries use the device casing itself for that.
"Regular" Lithium-Ion batteries are cylindrical metal cans, where instead of the polymer keeping the components pushed against eachother, the metal container exerts mechanical pressure on its contents.
Cylindrical cells are the most common type used for laptops, with the 18650 size being the most common. (18mm by 65mm). Yiur typical laptop battery has somewhere between 2-9 of these cells, in various parallell and series arrangements. A battery management, protection, and sometimes, "tell laptop this is genuine expensive battery"-chip complete the battery pack, which is then encased in hard plastic.
When a laptop battery fails, it's usually because one set cells in series arrangement has died. The entire pack is limited by its weakest participant. I recently disassembled a dead 6 cell battery. It was arranged as 2P3S, that is, 2 cells in parallell, 3 of these pairs in series. 1 pair was totally dead, had leaked, and its mechanical protection had severed its internal connections. Must've been running hot! Anither pair had so-so performance, and one pair had excellent performance.
Now, laptop battery recycling already takes place in China. Enterpresing people obtain dead laptop batteries, crack them open to harvest the cells that still function a little bit, and reuse them to make "new" laptop batteries. Thesd are sold on ebay, at significantly lower price compared to official genuine packs. The plastic casing, labels and holograms cost cents to make, and are not reused.
The cells nit reused for laptop packs, are given a new shrinkwrap, a fantasy capacity label, sometimes a circuit board is added, and then sold on. There's a significant market with e-cig users and fancy flashlight users, for cylindrical li-ion cells. It's remarkable that such a market exists, considering that no reputable manufacturer willingly or knowingly sell "bare" li-ion cells, thry always require the cells to be either built into a device, or buolt into a battery pack, and measures in place to prevent charging by anything except the designated charger. As such, one can consider all "lose" cells for sale a bit dubious.