If your computer isn't turning on, it could be that your battery or charger is deficient.
To determine if the charger is deficient, try plugging it in. When you do, check if the charging indicator on the front of the laptop lights up. If it does not, make sure the charging cable is receiving power: try plugging it in to a different outlet. If the laptop still does not get charged after this, you might want to consider getting a new charger.
If you are sure that your laptop charger is not broken, and you can turn the laptop on when it is plugged into the charger, try leaving the laptop on the charger of an extended period of time before unplugging it from the charger. If the laptop does not boot up after this, then your battery may be faulty. Consider replacing it.