I Dropped My Laptop & Now the Battery Won't Charge
Despite their portability, laptops aren't exactly designed to be thrown around. Developers and designers take the nature of the computer into account and try to plan for accidents. Some companies, Dell most notably, even offer warranties that protect laptops against falls and spills. But you can't prevent every accident, and sometimes dropping your laptop might lead to hardware malfunctions.
After dropping your laptop, have a close look at it to see if you can find any obvious physical damage: cracks in the plastic or loose parts, specifically. Serious physical damage will be plainly visible in most cases, but you may not always be able to see damage that might lead to the battery not holding a charge. In the event that the battery won't stay in the case, it's likely that the drop caused substantial damage to the battery bay, which can stop the battery from charging.
Identify Specific Area
Shut the computer off and unplug it and then remove the battery. Every laptop is a little different in terms of how the battery connects, so check your manual if you aren't sure how to remove your specific battery. Then look at the battery's connectors -- the gold contacts on one side -- for damage or corrosion. In rare circumstances, the shock from dropping a computer will cause the battery to rupture and leak, which could damage your computer. Check inside the battery bay for damage, too. If the area where the battery connects to the laptop is damaged, the battery won't charge.
Leave the battery disconnected and try running the laptop just on AC power. If the computer operates normally with the battery removed, you know that the fall didn't damage the motherboard or the connector pin for the external power pack. If the computer doesn't turn on with the battery removed, however, the AC connector pin is likely damaged, and it's possible the computer was running on battery power until exhausted.
The final troubleshooting step is to replace the battery or to try a backup. If you bought your computer with a backup or secondary battery, try it for a while in place of the battery that doesn't charge. Alternatively, you can contact the manufacturers and get a new battery. If your new battery doesn't charge either, the problem is with the power connections inside the laptop, not with the battery.