One of the most common issues with laptop batteries is overheating. Not that the batteries are designed poorly or improperly but due to, in most cases, user wear and tear. While the laptop is using the AC adapter for power the battery acts as a backup in case of a loss of power or accidental unplugging. During this AC power use the battery becomes hotter than it would if being used independently. To lessen the chances of this overheating, which leads to a shorter life expectancy of the battery, keep the AC power adapter plugged in only if you need to charge the battery while using the device.
If the battery is consistently hot to the touch, remove it and allow it to cool down making sure it is close to room temperature before reinserting it. Should the battery continue to overheat or get too hot it will probably need to be replaced. Always replace a defective battery as soon as possible as it could cause irreparable damage to the laptop in question. In addition to overheating, failure to charge or hold a charge is another very common issue with laptop batteries.
First check to see if the AC adapter is securely plugged into the device; the tightness of the connection will vary from laptop to laptop but it should be held in place well. To check for a good connection look for the battery charging LED if one is present on your laptop.
If it is not on wiggle the cord to see if the connection makes and breaks contact. If you notice a flickering LED light then the solution is a new adapter which is far better than needed to purchase a new battery or a new DC power jack which can be repaired by a common user with a little instruction and help.